Lectures on resonances

Johannes Sjöstrand^{∗}

1 Introduction

This text is the written version of lectures on resonances given at the Mathematics department of the Gothenburg University and Chalmers in the Spring semesters of 2000 to 2002. Some of the first chapters were also given in a graduate course at Ecole Polytechnique. Despite the somewhat preliminary form of these notes we hope they can be of some use as an introduction to the subject, and some guidance for a deeper study (necessarily involving the reading of research papers). We hope to improve and complete the text gradually.

Acknowledgements. I am grateful to the people at the Mathematics department of Chalmers and the Gothenburg University for giving me very nice working conditions, and the impulse to prepare this course. Thanks also to many other collegues and students at that department for stimulating discussions, and for making me feel well-come in Gothenburg. Thanks also to J.F. Bony for helping me to find a presentation of the material in Chapter 12, even though I finally settled for something less ambitious than what was originally planned. Last but not least, I would like to thank those who have courageously followed the lectures and showed their interest.

Palaiseau, 31.5.2002 the author

∗Centre de Mathématiques, Ecole Polytechnique, F-91128 Palaiseau cedex, UMR 7640, CNRS

2 Compactly supported perturbations of −Δ_{R}^{n}, definition of resonances and their multiplicity.

2.1 The free resolvent.

We start by recalling some facts about the “free Laplacian” P_{0} = −Δ = ∑_{1}^{n} −\frac{∂^{2}}{∂x_{j}^{2}} and its resolvent. First of all, −Δ is an unbounded selfadjoint operator L^{2}(R^{n}) → L^{2}(R^{n}) with domain H^{2}(R^{n}) (the standard Sobolev space). By Fourier transform it is unitarily equivalent to the (unbounded) multiplication operator L^{2}(R^{2}) ∋ û(ξ) ↦ ξ^{2}û(ξ) ∈ L^{2}(R^{n}). The spectrum of −Δ is equal to [0, +∞[ (and it is purely absolutely continuous).

For Im λ > 0, we know that λ^{2} ∉ [0, +∞[, so P_{0} − λ^{2} : H^{2}(R^{2}) → L^{2}(R^{n}) is bijective with inverse R_{0}(λ), which is a convolution operator with locally integrable kernel

R_{0}(λ; x − y) = \frac{i}{4}(\frac{1}{2π|x − y|})^{\frac{n−2}{2}} (u^{\frac{n−2}{2}} H(1)_{\frac{n−2}{2}} (u))_{|}_{u=λ|x−y|},  (2.1)

where H(1)_{\frac{n−2}{2}} is a Hankel function. We can deduce (see Vainberg [91]) that

R_{0}(λ) = \{ \begin{matrix}E(λ) + λ^{n−2}E_{1}(λ), n odd \\ F (λ) + F_{1}(λ)λ^{n−2} log λ, n even, \end{matrix}  (2.2)

where E(λ), E_{1}(λ), F (λ), F_{1}(λ) are entire functions with values in the space of convolution operators and in the space of classical pseudodifferential operators of order −2: H_{comp}(R^{n}) → H_{loc}^{2}(R^{n}).

If χ_{1}, χ_{2} ∈ C_{0}^{∞}(R^{n}) with supp χ_{1} ∩ χ_{2} = ∅, then χ_{1}(any of E, E_{1}, F, F_{1})χ_{2} is an entire function in λ with values in L(H^{−s}, H^{s}), for every s > 0.

Further for n even:

R_{0}(e^{imπ}z) = R_{0}(z) − (\frac{im}{4π})(−1)^{\frac{n−2}{2} (m+1)}T (z)  (2.3)

for 0 ≤ arg ≤ π (z ≠ 0,) m ∈ Z, and e^{imπ}z is viewed as a point on the logarithmic covering space of C \ {0}. Here T (z) is the convolution operator with kernel

T (z, x − y) = (\frac{z}{2π})^{n−2} ∫_{S^{n−1}} e^{iz(x−y)w}L(dw),  (2.4)

where L(dw) denotes the standard Lebesgue measure on S^{n−1}.

For n odd we have

R_{0}(z) − R_{0}(−z) = T (z),  (2.5)

where

T (z, x − y) = \frac{i}{2} \frac{z^{n−2}}{(2π)^{n−1}} ∫_{S^{n−1}} e^{iz(x−y)w}L(dw).  (2.6)

The basic facts to remember are that R_{0}(λ), initially defined for Im λ > 0 and viewed as a convolution operator H_{comp}^{0}(R^{n}) → H_{loc}^{2}(R^{n}), can be extended holomorphically to:

– λ ∈ C, when n is odd ≥ 3,

– λ ∈ C \ {0}, when n = 1, and the extension has a simple pole at λ = 0,

– to the logarithmic (universal) covering space of C \ {0}, when n is even. These properties have an interesting relation to the strong Huygens prin-

ciple for the wave equation. It is well known that the Cauchy problem in R × R^{n}:

(∂_{t}^{2} − Δ)U_{0}(t) = 0, U_{0}(0) = 0, U′_{0}(0) = δ,  (2.7)

has a unique solution U_{0} ∈ C^{∞}(R_{t}; D′(R_{x}^{n})) and that

supp U_{0} ⊂ {(t, x); |x| ≤ |t|}.

This is the Huygens principle which is valid more generally for wave equations with variable coefficients and more general hyperbolic equations, and which says that the support of a solution cannot travel faster than the natural wave propagation speed (which is one in our special case). When n is odd and ≥ 3, we have the strong Huygens principle:

supp U_{0} = {(t, x); |x| = |t|}

This principle has the important practical consequence that we can communicate and send intelligible signals using sound waves and electromagnetic waves, since we are lucky enough to live in a three dimensional world.

By Duhamel’s principle, we have the forward fundamental solution of the wave operator E_{0} ∈ S′(R × R^{n}) with

(∂_{t}^{2} − Δ)E_{0} = δ_{R^{n+1}}, E_{0} ⊂ {(t, x); t ≥ |x|},  (2.8)

given by

E_{0} = H(t)U_{0}(t),  (2.9)

where H(t) = 1_{[0,+∞[}(t) is the Heaviside function. Viewing U_{0} as a convolution operator, solving (∂^{2} − Δ)U_{0}(t) = 0, U_{0}(0) = 0, U′_{0}(0) = 1 (1 indicating the identity operator) we can write

U_{0}(t) = \frac{sin t\sqrt{−Δ}}{\sqrt{−Δ}}.  (2.10)

Without any general spectral theory, (2.10) simply says that for the Fourier transform with respect to x, we have

Û_{0}(t)(ξ) = \frac{sin t|ξ|}{|ξ|},

a fact that can be easily verified by taking the Fourier transform with respect to x of (2.8).

On the other hand, for Im λ > 0, it is easy to see that

R_{0}(λ) = ∫_{0}^{∞} e^{itλ}U_{0}(t)dt.  (2.11)

This will be verified in a more general situation in Chapter 3. The same formula holds on the level of convolution kernels. If n is odd ≥ 3, we see that U_{0}(t, x) = 0 for t > |x| by the strong Huygens principle, so in a region |x| < R we have

R_{0}(λ, x) = ∫_{0}^{R+1} e^{itλ}U_{0}(t, x)dt,  (2.12)

which can be extended holomorphically in λ to all of C.

Using (2.11), we can also derive explicit formulas for R_{0}, from corresponding explicit formulas for the fundamental solution. In odd dimensions the computations are easier. It is known (see volume 1 of [42]) that we have for t > 0:

⟨U_{0}(t), χ⟩ = \frac{1}{4π^{\frac{n−1}{2}}} (\frac{1}{2t} \frac{∂}{∂t})^{\frac{n−3}{2}} t^{n−2} ∫_{|ω|=1} χ(tω)L(dω),  (2.13)

where L(dω) denotes the Lebesgue measure, and n is odd and ≥ 3. Take n = 3 for simplicity, so that

⟨U_{0}(t), χ⟩ = \frac{1}{4π} t ∫_{|ω|=1} χ(tω)L(dω).  (2.14)

Then

⟨R_{0}(λ), χ⟩ = \frac{1}{4π} ∫_{|ω|=1} ∫_{0}^{∞} e^{itλ}χ(tω)tdtL(dω).  (2.15)

With x = tω, t = |x|, |ω| = 1, we have t^{2}dtdω = dx, so

⟨R_{0}(λ), χ⟩ = \frac{1}{4π} ∫_{R^{n}} \frac{1}{|x|}e^{i|x|λ}χ(x)dx,  (2.16)

or in other terms,

R_{0}(λ, x) = \frac{e^{i|x|λ}}{4π|x|}.  (2.17)

For higher odd values of n one gets a similar formula with the same exponential factor and a more complicated polynomial expression in front. For even n the expression is even more complicated and contains special functions. For n = 1 the computation is of course easy and is left as an exercise. In all cases, R_{0}(λ, x) is smooth and even analytic for x ≠ 0, locally integrable in x, and exponentially decaying (for λ in the upper halfplane) when x → ∞. These properties can also be deduced by means of Fourier inversion.

v

Here is a more general way to understand the link between the dimension and the extendability properties. Starting from the physical halfplane Im λ > 0, we use Fourier inversion to write

R_{0}(λ, x) = \frac{1}{(2π)^{n}} ∫ e^{ix·ξ} \frac{1}{ξ^{2} − λ^{2}} dξ.  (2.18)

To avoid uninteresting convergence problems, we take ϕ̂ ∈ C_{0}^{∞}(R^{n}), so that ϕ = F^{−1}ϕ̂ is of class S on R^{n} and extend to an entire function on C^{n}. In the sense of distributions, we get

⟨R_{0}(λ), ϕ̂⟩ = ∫ \frac{1}{(ξ^{2} − λ^{2})}ϕ(ξ)dξ.  (2.19)

Now pass to polar coordinates, but let r vary in R rather than in [0, ∞[, so that we get the double covering

R × S^{n−1} ∋ (r, ω) ↦ rω ∈ R^{n}.  (2.20)

Then the pull back of dξ becomes

|r|^{n−1}drdL(ω) = \{ \begin{matrix}r^{n−1}drL(dω) for n odd. \\ (sgn r)r^{n−1}drL(dω) for n even. \end{matrix}

We can then write

⟨R_{0}(λ), ϕ̂⟩ = \frac{1}{2} ∫_{S^{n−1}} ∫_{γ} \frac{ϕ(rω)}{r^{2} − λ^{2}} r^{n−1}drL(dω),  (2.21)

where γ is the positively oriented real axis when n is odd and the union of [0, −∞[ and [0, +∞[, both oriented from 0 to ∞, when n is even. Let λ start in the upper half plane and turn once around 0 in the clockwise direction. [[[[ Figur!]]]] From this simple argument we see that R_{0}(λ) extends to a holomorphic function on C \ {0} when n is odd, while we get a log type singularity when n is even. From this one can verify or at least understand qualitatively the earlier statements involving T (λ).

2.2 Definition of resonances in the case of P = −Δ + V, when V is bounded and compactly supported.

Let V ∈ L_{comp}^{∞}(R^{n}; R). Then P := P_{0} + V is an unbounded selfadjoint operator in L^{2}(R^{n}) with domain H^{2}. [0, +∞[ is the essential spectrum, so in ]−∞, 0[ there can only be isolated eigenvalues of finite multiplicity. We then know that the resolvent (P − λ^{2})^{−1} exists as a bounded operator H^{0} → H^{2}, when Im λ > 0, λ ≠ iμ_{j}, where −μ_{j}^{2} are the negative eigenvalues. We want to find a meromorphic extension of the resolvent as a continuous operator L_{comp}^{2}(R^{n}) → H_{comp}^{2}(R^{n}) across R \ {0}. Write

P − λ^{2} = (P_{0} − λ^{2}) + V = (1 + V R_{0}(λ))(P_{0} − λ^{2}).  (2.22)

For |λ| ≫ 1, arg λ = \frac{π}{4}, it is clear that (1 + V R_{0}(λ))^{−1} exists. Moreover V R_{0}(λ) : L^{2} → L^{2} is compact and depends holomorphically on λ for Im λ > 0. We conclude by analytic Fredholm theory (see Chapter 5) that (1 + V R_{0}(λ))^{−1} exists except for λ in some discrete set, where the inverse has poles of finite multiplicity, with finite rank coefficients in the singular terms of the Laurent expansions. (Actually these poles are precisely the iμ_{j}, because (2.22) shows that P − λ^{2} is bijective H^{2} → H^{0} precisely when 1 + V R_{0}(λ) is bijective.)

After crossing R \ {0}, V R_{0} is in general no more a bounded operator in L^{2}, but we can still try to solve

(1 + V R_{0}(λ))u = v  (2.23)

with u, v of compact support. Notice that the support of such a solution u will be contained in the union of the supports of v and V. Let 1_{supp V} ≺ χ ∈ C_{0}^{∞} where we write χ_{1} ≺ χ_{2} for two functions χ_{1}, χ_{2} if the support of χ_{1} is contained in the interior of the support of 1 − χ_{2}. Then 1 + V R_{0}(λ)χ is a holomorphic family of Fredholm operators (see Chapter 5) and we shall see that in order to solve (2.23) it suffices to invert 1 + V R_{0}(λ)χ.

Indeed, we have

1 + V R_{0}(λ) = (1 + V R_{0}(λ)(1 − χ))(1 + V R_{0}(λ)χ),  (2.24)
1 = (1 + V R_{0}(λ)(1 − χ))(1 − V R_{0}(1 − χ)),  (2.25)

so

(1 + V R_{0}(λ)(1 − χ))^{−1} = 1 − V R_{0}(λ)(1 − χ) on L_{comp}^{2}.  (2.26)

Hence (1+V R_{0}(λ)) : L_{comp}^{2} → L_{comp}^{2} is invertible precisely when 1+V R_{0}(λ)χ is, and

(1 + V R_{0}(λ))^{−1} = (1 + V R_{0}(λ)χ)^{−1}(1 − V R_{0}(λ)(1 − χ)) : L_{comp}^{2} → L_{comp}^{2}.

(2.27)

Inverting (2.22), we get

(P − λ^{2})^{−1} = R_{0}(λ)(1 + V R_{0}(λ))^{−1},  (2.28)

and by analytic Fredholm theory (see chapter 5), we know that (1+V R_{0}(λ)(1− χ))^{−1} has a meromorphic extension in λ to the complex plane when n ≥ 3 is odd and to other sets already described in the other cases, as a function with values in the space L(L^{2}(R^{n}), L^{2}(R^{n})) of bounded operators from L^{2} to L^{2}. By a meromorphic operator valued function of z ∈ Ω, with values in L(H_{1}, H_{2}), where H_{j} are complex Hilbert spaces and Ω is an open set in the complex plane or in a covering surface over some open set of C, we mean a function A(z) which is holomorphic in z on Ω \ S, where S is a discrete subset of Ω and such that if z_{0} ∈ S, then for z in a neighborhood of z_{0}, we have the Laurent series representation

A(z) = ∑_{1}^{N} \frac{A_{j}}{(z − z_{0})^{j}} + B(z),  (2.29)

where N is finite (so that we have poles of finite order), B(z) a holomorphic operator valued function in a neighborhood of z_{0}, and A_{j} are bounded operators, independent of z and of finite rank. We leave as an exercise to verify that compositions and sums of meromorphic operator valaued functions are still meromorphic. Meromorphic operator valued functiosn are also conserved under changes of variables z = κ(λ) if κ : Λ → Ω is a biholomorphic map beteween two open sets. Proposition 2.1 (P − λ^{2})^{−1} : H_{comp}^{0} → H_{loc}^{2} has a meromorphic extension from the open upper half plane to: C \ {0}, when n = 1, C, when n ≥ 3 is odd, the logarithmic covering space (C \ {0})^{∗} of C \ {0}, when n is even.

By definition, the resonances or scattering poles of P are the poles of this extension, with the exception of the iμ_{j}. Since H_{comp}^{0}, H_{loc}^{2} are not Hilbert or even Banach spaces, we have to extend the definition of meromorphic and holomorphic functions with values in operators between such spaces. Let Ω be as above. A holomorphic function with values in L(H_{comp}^{0}, H_{loc}^{2}) is a function A(z) with values in the space of linear operators H_{comp}^{0} → H_{loc}^{2}, such that χ_{1}A(z)χ_{2} is holomorphic for all χ_{j} ∈ C_{0}^{∞}(R^{n}). Correspondingly a meromorphic function is one which is holomorphic on Ω \ S, where S ⊂ Ω is discrete, and such that if z_{0} ∈ S, then near z_{0}, we have (2.29), now with A_{j} : H_{comp}^{0} → H_{loc}^{2} (continuous in the sense that χ_{1}A_{j}χ_{2} is bounded for all χ_{j} as above) of finite rank, and B(z) holomorphic with values in L(H_{comp}^{0}, H_{loc}^{2}) for z in a neighborhood of z_{0}.

2.3 Black box frame work and definition of resonances for general compactly supported perturbations of −Δ.

The black box frame work (introduced by Sjöstrand and Zworski in [80]) permits to give a uniform treatment of all sorts of compactly supported (and later also long range) perturbations of −Δ, like exterior obstacle problems, metric pertubations, Schrödinger operators, or a combination of such problems.

Let H be a complex Hilbert space with the orthogonal decomposition

H = H_{R_{0}} ⊕ L^{2}(R^{n} \ B(0, R_{0})),  (2.30)

where R_{0} > 0 and we use the standard notation B(x, r) = {y ∈ R^{n}; |y −x| < r}. We denote by

u ↦ u_{|}_{B(0,R_{0})} or 1_{B(0,R_{0}}u

u ↦ u_{|}_{R^{n}\B(0,R_{0})} or 1_{R^{n}\B(0,R_{0})}u,

the corresponding orthogonal projections. If χ is a bounded continuous function on B(0, R_{0}); χ ∈ C_{b}(R^{n}), and χ = C_{0} = Const. on B(0, R_{0}), then for u ∈ H, we can define χu ∈ H by

χu = C_{0}u + 1_{R^{n}\B(0,R_{0})}(χ − C_{0})u.  (2.31)

Let

H_{comp} = {u ∈ H; u_{|}_{R^{n}\B(0,R_{0})} has bounded support},
H_{loc} = H_{R_{0}} ⊕ L_{loc}^{2}(R^{n} \ B(0, R_{0})).

Notice that if χ ∈ C(R^{n}) is constant near \overline{B(0, R_{0})}, then u ↦ χu is well defined: H_{comp/loc} → H_{comp/loc}.

We now consider a selfadjoint unbounded operator P : H → H with domain D ⊂ H. We assume

u ∈ D ⇒ u_{|}_{R^{n}\B(0,R_{0})} ∈ H^{2}(R^{n} \ B(0, R_{0})),  (2.32)

and that the restriction operator is continuous,

u ∈ H^{2}(R^{n} \ B(0, R_{0})), u = 0 near ∂B(0, R_{0}) ⇒ u ∈ D,  (2.33)

where as often we identify H^{0}(R^{n} \ B(0, R_{0})) with a subspace of H in the natural way,

(P u)_{|}_{R^{n}\B(0,R_{0})} = −Δ(u_{|}_{R^{n}\B(0,R_{0})}), ∀u ∈ D.  (2.34)

Notice that if u is as in (2.33) (with vanishing H_{R_{0}} component) then P u = −Δu (i.e. the H_{R_{0}} component of P u vanishes) since for every v ∈ D:

(P u|v) = (u|P v) = (u| − Δ(v_{|}_{R^{n}\B(0,R_{0})})) = (−Δu|v)_{L^{2}(R^{n}\B)}.

We also assume that

1_{B(0,R_{0})}(i − P)^{−1} is compact: H → H.  (2.35)

Define D_{comp} = D∩H_{comp}, D_{loc} = {u ∈ H_{loc}; χu ∈ D, ∀χ ∈ C_{0}^{∞}(R^{n}) with χ = Const. near \overline{B(0, R_{0})}\}.

We should keep i mind the following two examples:

1) Let P = −Δ+V (x), where V ∈ L_{comp}^{∞}(R^{n}). Then we can take H = L^{2}(R^{n}) and the domain of our self-adjoint operator P is D = H^{2}(R^{n}).

2) Let O ⊂ R^{n} be a bounded open set with smooth boundary. Let P be the unbounded self-adjoint realization of −Δ on L^{2}(R^{n} \ O) with domain D = {u ∈ H^{2}(R^{n} \ O); u_{|}_{∂O} = 0}. P is often called the exterior Dirichlet Laplacian. Similarly we can define the exterior Neumann Laplacian by taking D = {u ∈ H^{2}(R^{n}\O); ∂_{ν}u_{|}_{∂O} = 0}, where ∂_{ν}u denotes the exterior derivative in the unit normal direction.

Theorem 2.2 a) P has only discrete spectrum in ] − ∞, 0[,

b) (P − λ^{2})^{−1} : H_{comp} → D_{loc} has a meromorphic extension from {λ ∈ C; Im λ > 0, λ^{2} ∉ σ(P)∩] − ∞, 0[} to: C \ {0} when n = 1, C when n ≥ 3 is odd, (C \ {0})^{∗} when n is even.

Here meromorphic functions with values in the space of continuous operators H_{comp} → D_{loc} are defined in the same way as in the special case of such functions with values in the continuous operators: H_{comp}^{0} → H_{loc}^{2}. Proof. We first “enlarge” the compactness property (2.35). If z, w ∈ ρ(P), the resolvent set of P, then we have the resolvent identity

(z − P)^{−1} = (w − P)^{−1} − (w − P)^{−1}(z − w)(z − P)^{−1},  (2.36)

implying that

1_{B(0,R_{0})}(z−P)^{−1} = 1_{B(0,R_{0})}(i−P)^{−1}−1_{B(0,R_{0})}(i−P)^{−1}(i−w)(z−P)^{−1} (2.37)

is compact. Passing to the adjoints, we see that (z − P)^{−1}1_{B(0,R_{0})} is compact too.

Let B(x, r_{0}, r_{1}) denote the shell {y ∈ R^{n}; r_{0} < |y − x| < r_{1}}. Then 1_{B(0,R_{0},R)} : H^{2}(R^{n} \ B(0, R_{0})) → L^{2}(R^{n} \ B(0, R_{0})) is compact for R_{0} < R < ∞. It follows that

1_{B(0,R)}(z − P)^{−1}, (z − P)^{−1}1_{B(0,R)} : H → H  (2.38)
are compact for z ∈ ρ(P), R_{0} < R < ∞.

Let χ_{0}, χ_{1}, χ_{2} ∈ C_{0}^{∞} with 1_{B(0,R_{0})} ≺ χ_{0} ≺ χ_{1} ≺ χ_{2}. We first let λ ∈ C with Im λ > 0 and take μ with the same properties and such that μ^{2} ∉ σ(P). We will fix μ later. Put

Q(λ, μ) = (1−χ_{0})R_{0}(λ)(1−χ_{1})+χ_{2}R(μ)χ_{1} = Q_{0}(λ, μ)+Q_{1}(λ, μ) : H → D, where R_{0}(λ) = (P_{0} − λ^{2})^{−1}, R(μ) = (P − μ^{2})^{−1}, and the last equality above gives the obvious definition of Q_{0}, Q_{1}. Then:

(P − λ^{2})Q_{0} = (1 − χ_{1}) + [Δ, χ_{0}]R_{0}(λ)(1 − χ_{1}) = (1 − χ_{1}) + K_{0}(λ), (2.39)

(P − λ^{2})Q_{1} = χ_{2}(P − λ^{2})R(μ)χ_{1} − [Δ, χ_{0}]R_{0}(μ)χ_{1}  (2.40)
= χ_{1} + χ_{2}(μ^{2} − λ^{2})R(μ)χ_{1} − [Δ, χ_{2}]R(μ)χ_{1} = χ_{1} + K_{1}(λ, μ),

where the last equations define the operators K_{0}, K_{1}. Hence,

(P − λ^{2})Q = 1 + K_{0}(λ) + K_{1}(λ, μ) = 1 + K(λ, μ),  (2.41)

where the last equality defines K.

Using (2.38) and the fact that [Δ, χ_{j}] is a first order operator with smooth compactly supported coefficients and therefore compact: H^{2}(R^{n}\B(0, R_{0})) → L^{2}(R^{n} \ B(0, R_{0})), we see that

K(λ, μ) is compact.  (2.42)

Now choose μ with arg μ = \frac{π}{4}, so that Im (μ^{2}) = |μ|^{2}. Then

R(μ) = \{ \begin{matrix}O(\frac{1}{|μ|^{2}} ) : H → H, \\ O(1) : H → D, \end{matrix}

and,

(1 − χ_{0})R(μ) = \{ \begin{matrix}O(\frac{1}{|μ|^{2}} ) : H → L^{2} \\ O(1) : H → H^{2}, \end{matrix}

so by interpolation, (1 − χ_{0})R(μ) = O(\frac{1}{|μ|}) : H → H^{1}. It follows that

K(μ, μ) = O(\frac{1}{|μ|}) : H → H,  (2.43)

and consequently 1 + K(μ, μ) : H → H is invertible with a bounded inverse (1 + K(μ, μ))^{−1} of norm ≤ 2 when |μ| is large enough (and arg μ = π/4). We fix such a value of μ and apply analytic Fredholm theory with respect to λ. It follows that (1 + K(λ, μ))^{−1} exists except for λ in a discrete set in the open upper half plane, and that the poles are of finite order and with coefficients of finite rank for the singular terms in the Laurent expansion at each such pole. (See chapter 5.)

Writing

(P − λ^{2})^{−1} = Q(λ, μ)(1 + K(λ, μ))^{−1},  (2.44)

we see that (P − λ^{2})^{−1} : H → D is meromorphic in the upper half plane with only finite rank singular terms in the Laurent expansion. Of course, we already know that the only possible singularities are situated on the negative imaginary axis and correspond to the negative spectrum of P, so we see that that this spectrum is purely discrete and we have obtained part (a) of the theorem. (The spectral projection onto any finite part of the spectrum is given by \frac{1}{2πi} ∫_{γ}(λ^{2} − P)^{−1}d(λ^{2}), where γ is simple closed contour in the upper half plane which avoids the poles and encircles a certain number of square roots of negative eigenvalues.)

We next consider the meromophic extension and let n ≥ 3 be odd in order to fix the ideas. We keep μ fixed as before. Then C ∋ λ ↦ Q(λ, μ) : H_{comp} → D_{loc} is holomorphic. Let χ_{2} ≺ χ ∈ C_{0}^{∞}(R^{n}), so that χK = K. To invert 1 + K : H_{comp} → H_{comp} it suffices to invert 1 + Kχ : H → H. Indeed, we observe that (1 − χ)K = 0 and hence

(1 + K(1 − χ))^{−1} = 1 − K(1 − χ), (1 + K) = (1 + K(1 − χ))(1 + Kχ),

implying

(1 + K)^{−1} = (1 + Kχ)^{−1}(1 − K(1 − χ)).  (2.45)

Using (2.45) in (2.44) and applying analytic Fredholm theory to (1 + Kχ)^{−1}, we get the desired meromorphic extension

R(λ) = Q(λ, μ)(1 + K)^{−1} = Q(1 + Kχ)^{−1}(1 − K(1 − χ)).  (2.46)

#

In the following we shall sometimes use the notation R(λ) = (P − λ)^{−1} for the meromorphic extension of the resolvent, even though it is no more an inverse, strictly speaking. The poles of the meromorphic extension not already in the physical halfplane C_{+} := {λ ∈ C; Im λ > 0} that was the starting point of the extension, will be called resonances (and sometimes scattering poles). If λ_{0} is a non-vanishing resonance, we define its multiplicity m(λ_{0}) by

m(λ_{0}) = rank \frac{1}{2πi} ∫_{γ}(λ^{2} − P)^{−1}d(λ^{2}),  (2.47)

where γ is a small positively oriented circle centered at λ_{0}. With z_{0} = λ_{0}^{2}, z = λ^{2}, we have for z close to z_{0}:

(z − P)^{−1} = (z − z_{0})^{−N}Π_{−N} + .. + (z − z_{0})^{−1}Π_{−1} + Hol (z) : H_{comp} → D_{loc} ,

(2.48)

where the last term is holomorphic in a neighborhood of z = z_{0}. From this expression we see that m(λ_{0}) is the rank of Π_{−1} in view of the standard formula:

Π_{−j} = \frac{1}{2πi} ∫_{γ}(z − z_{0})^{j−1}(z − P)^{−1}dz, j ≥ 1.  (2.49)

Here γ̃ denotes the contour in the z plane which corresponds to γ in the λ plane.

Use that

(P − z_{0})(z − P)^{−1} = (z − z_{0})(z − P)^{−1} − 1

by holomorphic extension from the physical sheet, to get for j ≥ 1:

(P − z_{0})Π_{−j} = \frac{1}{2πi} ∫_{γ}(z − z_{0})^{j−1}(P − z_{0})(z − P)^{−1}dz (2.50)

= \frac{1}{2πi} ∫_{γ}(z − z_{0})^{j}(z − P)^{−1}dz − \frac{1}{2πi} ∫_{γ}(z − z_{0})^{j−1}dz = Π_{−(j+1)}

Consequently (P − z_{0}) maps Π_{−j}(H_{comp}) into Π_{−(j+1)}(H_{comp}) (also included in D_{loc}). Since Π_{−j} are of finite rank and D_{comp} is dense in H_{comp} [[détailler cet argument]], we have Π_{−j}(H_{comp}) = Π_{−j}(D_{comp}). Moreover, on D_{comp} we have Π_{−j}P = P Π_{−j}, so (2.50) becomes:

Π_{−(j+1)} = Π_{−j}(P − z_{0}) on D_{comp}.  (2.51)

Consequently,

Π_{−(j+1)}(H_{comp}) = Π_{−(j+1)}(D_{comp}) ⊂ Π_{−j}(H_{comp}),

and (P − z_{0})_{|}_{Π_{−1}(H_{comp})} is a nilpotent operator. The elements of the kernel of this operator will be called resonant states. The elements of Π_{−1}(H_{comp}) will be called generalized resonant states.

If Π_{−1}v is a resonant state, then 0 = Π_{−2}v = Π_{−3}v = .. and (z − P)^{−1}v has a simple pole at z_{0}.

Proposition 2.3 Let 1_{B(0,R_{0})} ≺ χ ∈ C_{0}^{∞}(R^{n}). Then with γ as in (2.47),

m(λ_{0}) = rank ∫_{γ}(λ^{2} − P)^{−1}χd(λ^{2})  (2.52)

Proof. Clearly m(λ_{0}) is at least as large as the RHS of (2.52). Let χ̃ ∈ C_{0}^{∞}(R^{n}), with 1_{B(0,R_{0})} ≺ χ̃ ≺ χ. From (2.44), we get

(P − λ^{2})^{−1} = Q − (P − λ^{2})^{−1}K(λ, μ) = Q − (P − λ^{2})^{−1}χ̃K(λ, μ),

provided we let the cutoffs χ_{1}, χ_{2}, χ_{3} have their supports sufficiently close to B(0, R_{0}). Hence

\frac{1}{2πi} ∫_{γ}(λ^{2} − P)^{−1}d(λ^{2}) = −\frac{1}{2πi} ∫_{γ}(λ^{2} − P)^{−1}χ̃K(λ, μ)d(λ^{2}).  (2.53)

Recall that μ is fixed and write the Taylor expansion of K at λ_{0}^{2} in the variable λ^{2}:

K(λ, μ) = ∑_{j=0}^{N−1} K_{j}(μ)(λ^{2} − λ_{0}^{2})^{j} + (λ^{2} − λ_{0}^{2})^{N})Hol (λ),

where Hol (λ) is function which is holomorphic for λ close to λ_{0}. Use this in (2.53) together with (2.49):

\frac{1}{2πi} ∫_{γ}(λ^{2} − P)^{−1}d(λ^{2}) =  (2.54)

− ∑_{0}^{N−1} \frac{1}{2πi} ∫_{γ}(λ^{2} − λ_{0}^{2})^{j}(λ^{2} − P)^{−1}d(λ^{2})χ̃K_{j} = − ∑_{0}^{N−1} Π_{−(1+j)}χ̃K_{j}.

Here we use (2.50):

Π_{−(j+1)}χ̃ = Π_{−j}(P − z_{0})χ̃ = Π_{−j}χ̂(P − z_{0})χ̃,

if χ̃ ≺ χ̂, so

R(Π_{−(j+1)}χ̃) = R(Π_{−j}χ̂).

Iterating, we get

R(Π_{−(j+1})χ̃) ⊂ R(Π_{−1}χ),

and using this in (2.54), we get

R(Π_{−1}) ⊂ R(Π_{−1}χ),

which implies (2.52). #

2.4 Absence of real resonances.

We end this chapter by discussing the relation with outgoing solutions to the free Helmholtz equation, and then applying it to show that real non-vanishing resonances will not appear in standard situations. (Rellich’s theorem.) We shall first see that resonant states associeted to such resonances are outgoing states in a sense that we define below.

Let 1_{B(0,R_{0})} ≺ χ ∈ C_{0}^{∞}(R^{n}). For z = λ^{2}, Im λ > 0, we have for v ∈ H_{comp} with 1_{supp v} ≺ χ:

(z − P)^{−1}v = χ(z − P)^{−1}v + (z − P_{0})^{−1}[P, χ](z − P)^{−1}v,  (2.55)

where we notice that the second term of the RHS vanishes outside supp (1 − χ). In fact, let u = (z − P)^{−1}v and compute

(z − P_{0})((1 − χ)u) = (z − P)((1 − χ)u) = [P, χ]u

which implies that

(1 − χ)(z − P)^{−1}v = (z − P_{0})^{−1}[P, χ](z − P)^{−1}v.

The relation (2.55) extends by analytic continuation to the non-physical sheet(s). Let z_{0} = λ_{0}^{2} where λ_{0} is a resonance. Integrating (2.55) along a small contour γ around z_{0}, we get for z ∈ R^{n} \ \overline{B(0, R_{0})}:

Π_{−1}v − χΠ_{−1}v = ∑_{j=1}^{N} \frac{1}{2πi} ∫ (z − P_{0})^{−1}[P, χ](z − z_{0})^{−j}Π_{−j}vdz (2.56)

= ∑_{j=1}^{N} \frac{1}{(j − 1)!}(∂_{z}^{j−1}(z − P_{0})^{−1})_{z=z_{0}}[P, χ]Π_{−j}v.

Choose v ∈ H_{comp} so that Π_{−1}v is a resonant state. Then (2.56) simplifies to

Π_{−1}v = χΠ_{−1}v + (z_{0} − P_{0})^{−1}[P, χ]Π_{−1}v,  (2.57)

where we recall that (z_{0} − P_{0})^{−1} denotes the holomorphic extension from the physical sheet to the point z_{0}.

For simplicity, we now restrict the attention to a real positive z_{0} = z or equivalently to a real non-vanishing λ. Let R_{0}(λ) = (P_{0} − λ^{2})^{−1} be the branch of the resolvent obtained by extension from the upper half plane: R_{0}(λ) = (P_{0} − (λ + i0)^{2})^{−1} : E′ → S′.

Definition. We say that u ∈ S′(R^{n}) is outgoing if there exists w ∈ E′(R^{n}) such that

u(x) = R_{0}(λ)w for |x| ≫ 1.  (2.58)

We have just seen that a resonant state (or rather it’s component in L_{loc}^{2}(R^{n} \ B(0, R_{0}))) is outgoing.

Exercise: Let u ∈ S(R^{n}) be outgoing, so that u(x) = R_{0}(λ)w(x), |x| ≫ 1, for some w ∈ E′. Then (P_{0} − λ^{2})u = v ∈ E′. Show that u = R_{0}(λ)v.

Solution: As in (2.55) we have for χ ∈ C_{0}^{∞} with χ = 1 on a sufficiently large ball:

R_{0}(λ)w = χR_{0}(λ)w − R_{0}(λ)[P_{0}, χ]R_{0}(λ)w,

and hence u = −R_{0}(λ)[P_{0}, χ]u near ∞.

On the other hand, let 1_{supp v} ≺ χ. Then

(P_{0} − λ^{2})χu = v + [P_{0}, χ]u ∈ E′,  (2.59)

and on E′, we have R_{0}(λ)(P_{0}−λ^{2}) = 1 by analytic extension from the physical half plane. So applying R_{0}(λ) to (2.59), we get

χu = R_{0}(λ)v + R_{0}(λ)[P_{0}, χ]u.

Hence

R_{0}(λ)v = −R_{0}(λ)[P_{0}, χ]u = u near ∞.

u − R_{0}(λ)v is therefore in E′ and since it satisfies (P_{0} − λ^{2})(u − R_{0}(λ)v) = 0, it has to be 0. (Notice or recall that every non-vanishing partial differential operator with constant coefficients is injective E′(R^{n}) → E′(R^{n}).) #

Theorem 2.4 If λ ∈ R \ {0} is a resonance and u ∈ D_{loc} a corresponding resonant state, then u has compact support.

We notice that in most standard situations, there cannot be any nontrivial solution of the equation (P − λ^{2})u which has compact support and hence the theorem implies that that there are no real non-vanishing resonances. For the proof of the theorem we need Proposition 2.5 Let v ∈ C_{0}^{∞}(R^{n}), 1_{supp v} ≺ χ ∈ C_{0}^{∞}(R^{n}; R), 0 ≠ λ ∈ R. Then

\frac{1}{2i}([−Δ, χ]R_{0}(λ)v|R_{0}(λ)v) = C(λ) ∫_{∂B(0,|λ|)} |v̂(ξ)|^{2}L(dξ) = Im (R_{0}(λ)v|v),

(2.60)

where C(λ) ≠ 0 can be computed, and L(dξ) denotes the standard Lebesgue measure on the sphere of radius |λ|.

Proof. This is a simple computation, where we assume that λ > 0 for simplicity:

\frac{1}{2i}([−Δ, χ]R_{0}(λ)v|R_{0}(λ)v) = \frac{1}{2i}([−Δ − λ^{2}, χ]R_{0}(λ)v|R_{0}(λ)v)

= \frac{1}{2i}((χR_{0}(λ)v|(−Δ − λ^{2})R_{0}(λ)v) − (χ(−Δ − λ^{2})R_{0}(λ)v|R_{0}(λ)v))

= \frac{1}{2i}((R_{0}(λ)v|χv) − (χv|R_{0}(λ)v))

= Im (R_{0}(λ)v|χv) = Im (R_{0}(λ)v|v) =

Im \frac{1}{(2π)^{n}} ∫_{R^{n}} \frac{v̂(ξ)\overline{v̂}(ξ)}{(ξ^{2} − (λ + i0)^{2})}dξ = \frac{1}{(2π)^{n}} ∫_{R^{n}} |v̂(ξ)|^{2} Im \frac{1}{(|ξ|^{2} − (λ + i0)^{2})}dξ

= \frac{π}{(2π)^{n}2λ} ∫_{∂B(0,λ)} |v̂(ξ)|^{2}L(dξ).

#

Notice that by density the first equality in (2.60) holds also for v ∈ E′.

Proof of Theorem 2.4. Let u be an outgoing solution of (P − λ^{2})u = 0 with λ ∈ R_{+}. Let 1\overline{B(0,R_{0})} ≺ χ ∈ C_{0}^{∞} be real-valued and consider

\frac{1}{2i}([P, χ]u|u) = \frac{1}{2i}((χu|P u) − (χP u|u))

= \frac{1}{2i}(λ^{2}(χu|u) − λ^{2}(χu|u)) = 0.

On the other hand, if we write u = R_{0}(λ)v, v ∈ C_{0}^{∞}, for |x| ≫ 1, (we can take v = −[P_{0}, χ̃]u for a suitable χ̃), then if we let χ ∈ C_{0}^{∞} be a real-valued cut-off which is equal to 1 on a sufficiently large ball, we have

0 = (\frac{1}{2i}[P, χ]u|u) = \frac{1}{2i}([−Δ, χ]R_{0,+}(λ)v|R_{0,+}(λ)v)

= C(λ) ∫_{∂B(0,λ)} |v̂(ξ)|^{2}L(dξ), so v̂(ξ) = 0 on B(0, λ) and hence also on the complexification of this ball. By the Paley-Wiener theorem plus easy division estimates (see Lemma 2.6 below), we see that R_{0,+}(λ)v ∈ E′. Hence u has compact support. #

Remark. In many situations, we know that there are no non-trivial solutions of (P − λ^{2})u = 0 with compact support, and it then follows from Theorem 2.4 that there are no real non-vanishing resonances.

Lemma 2.6 Let v ∈ E′(R^{n}) with v̂(ξ) = 0 on the real sphere ξ^{2} = λ^{2}, for some λ > 0. Then R_{0}(λ)v ∈ E′(R^{n}) (which implies that supp R_{0}(λ)v is contained in the union of the support of v and the bounded components of the complement of the support of v).

Proof. We recall a weak form of the Paley-Wiener Theorem: If u ∈ S′(R^{n}), then u belongs to E′ if and only if

∃C_{0}, C_{1}, N_{0} > 0, such that û(ξ) is entire and  (2.61)
|û(ξ)| ≤ C_{1}⟨ξ⟩^{N_{0}}e^{C_{0}|Im ξ|}, ξ ∈ C^{n}.

If v satisfies the assumptions of the lemma, then û := \frac{_{v}(ξ)}{(ξ^{2}−λ^{2})} is an entire function and it is easy to see that û is the Fourier transform of R_{0}(λ)v. It suffices, then to show that û(ξ) satisfies (2.61) for some suitable constants.

Let δ > 0 be small (to be fixed later) and let ξ_{0} ∈ C^{n} with |ξ_{0}^{2} − λ^{2}| ≤ δ. Write

ξ^{2} − λ^{2} = 2ξ_{0}(ξ − ξ_{0}) + (ξ_{0}^{2} − λ^{2}) + (ξ − ξ_{0})^{2}.

Take

ξ − ξ_{0} = z \frac{\overline{ξ}_{0}}{|ξ_{0}|}, with z ∈ C, |z| = ϵ > 0,

where ϵ > 0 will be fixed independently of ξ_{0}.

Then

|ξ^{2} − λ^{2}| ≥ 2|ξ_{0}|ϵ − δ − ϵ^{2}.

λ being fixed, we first choose δ ≤ δ_{1}(λ) small enough, so that |ξ_{0}| ≥ \frac{|λ|}{2} and get

|ξ^{2} − λ^{2}| ≥ |λ|ϵ − δ − ϵ^{2}.

Choose ϵ < |λ|/2 so that |ξ^{2} − λ^{2}| ≥ |λ|ϵ/2 − δ and finally δ ≤ δ_{2}(λ, ϵ) so that |ξ^{2} − λ^{2}| ≥ |λ|ϵ/4. Consider

û(ξ_{0} + z \frac{\overline{ξ}_{0}}{|ξ_{0}|}) = \frac{v̂(ξ_{0} + z \frac{\overline{ξ}_{0}}{|ξ_{0}|)})}{(ξ_{0} + z \frac{\overline{ξ}_{0}}{|ξ_{0}|})^{2} − λ^{2}}

which is holomorphic for |z| < 2ϵ. We have

|û(ξ_{0} + z \frac{\overline{ξ}_{0}}{|ξ_{0}|})| ≤ 4\frac{|v̂(ξ_{0} + z \frac{\overline{ξ}_{0}}{|ξ_{0}|})|}{|λ|ϵ}

for |z| = ϵ, so by the maximum principle we get for |z| ≤ ϵ:

|û(ξ_{0} + z \frac{\overline{ξ}_{0}}{|ξ_{0}|})| ≤ \frac{4}{|λ|ϵ} sup_{|z|=ϵ} |v̂(ξ_{0} + z \frac{\overline{ξ}_{0}}{|ξ_{0}|})|.

Especially we get the last estimate for û(ξ_{0}), and we conclude that if v̂ satisfies (2.61), then û also satisfies (2.61) in the region |ξ^{2} − λ^{2}| ≤ δ, and hence everywhere, since |û(ξ)| ≤ δ^{−1}|v̂(ξ)| in the region where |ξ^{2} − λ^{2}| ≥ δ. #

Remark. It is a little easier to show that if

e_{0} ∈ D, (P − λ^{2})e_{0} = 0, λ ∈ R \ {0},

so that λ^{2} is a positive eigenvalue, then e_{0} has compact support. Take χ as usual, equal to one near \overline{B(0, R_{0})}. Then

(P_{0} − λ^{2})(1 − χ)e_{0} = [Δ, χ]e_{0} ∈ H_{comp}^{1},

so if F denotes the Fourier transformation, then

(ξ^{2} − λ^{2})F((1 − χ)e_{0})(ξ) = (F[Δ, χ]e_{0})(ξ).

Here the right hand side is entire and in order to have F(1 − χ)e_{0} ∈ L^{2}, we must have F[Δ, χ]e_{0})(ξ) = 0 when ξ^{2} = λ^{2}. As before, we conclude by means of Paley-Wiener, that

(F((1 − χ)e_{0})(ξ) = \frac{F([Δ, χ]e_{0})(ξ))}{ξ^{2} − λ^{2}}

is an entire function and satisfies (2.61). Hence (1−χ)e_{0} has compact support and so does e_{0}.

[[[Missing piece in this presentation: Show that if (P − λ^{2})u = 0 and u is outgoing, then u is a resonant state and λ is a resonance.]]]

3 Expansion of solutions to the wave equation in exponentially decaying resonant modes, when t → +∞.

In this chapter we show (in the case of odd dimension n ≥ 3) that if a certain non-trapping assumption for the associated wave equation is fulfilled, then for every N > 0 there are at most finitely many resonances above the curve

Im z = −N log_{+} |Re z|

(This fact is true also in even dimensions, but a little more difficult to prove in that case.) Moreover if the space dimension is odd and ≥ 3 we shall show that the solution of the associated wave equation with compactly supported initial data has an asymptotic expansion in exponentially decaying modes: e^{−iλ_{j}t}u_{j}(x, t) as t → +∞, where λ_{j} is a resonance or a square root of a negative eigen-value and u_{j} is a polynomial of degree ≤ ω(λ_{j}) − 1 in t, where ω(λ_{j}) is the order of λ_{j} as a pole of the meromorohic extension of (P −λ^{2})^{−1}. This result is due to Lax-Phillips (see [50]) in a more restricted setting, and the idea of the proof we give is due to Vainberg [91]. Vainberg’s proof was recently adapted to the black box setting by Tang and Zworski [90], but our proof (and result) differs slightly. At one point we shall use a basic result on the propagation of singularities for the wave operator. See [42].

Let P be a bb operator as in the preceding chapter, satisfying the additional assumption:

P ≥ −C,  (3.1)

for some constant C. Let D^{j} denote the domain of ⟨P ⟩^{j} when j ≥ 0 and let D^{j} = (D^{−j})^{∗} be the dual space of D^{−j}, when j ≤ 0.

Recall that by the spectral theorem, P is unitarily equivalent to the multiplication operator P̃ : u(m) ↦ p(m)u(m) in H̃ = L^{2}(M, μ) where M is some set, μ a measure on M and p(m) a real-valued measurable function on M. Then D^{j} becomes {⟨p(m)⟩^{−j}u(m); u ∈ L^{2}(M; μ)}. From this representation and the Hölder inequality, we deduce the interpolation inequality

∥u∥_{D^{θj+(1−θ)k}} ≤ ∥u∥_{D^{j}}^{θ}∥u∥_{D^{k}}^{1−θ}, j, k ∈ R, 0 ≤ θ ≤ 1, u ∈ D^{max(j,k)}. (3.2)

For g ∈ D^{j}, we can solve the abstract Cauchy problem

(P + ∂_{t}^{2})u = 0, u(0) = 0, ∂_{t}u(0) = g,  (3.3)

by the functional formula:

u(t) = \frac{sin t\sqrt{P}}{\sqrt{P}} g =: U(t)g.  (3.4)

(If we apply the unitary equivalence above, then U(t) can be identified with

the multiplication operator u(m) ↦ ^{\frac{sin(t\sqrt{p(m)})}{\sqrt{p(m)}}}u(m).) Note that f(t, E) :=

\frac{sin t\sqrt{E}}{\sqrt{E}} is a well-defined smooth function on R × R, independent of the choice of branch of \sqrt{E}, which satisfies the following estimates for E ≥ −Const.:

∂_{t}^{k}f(t, E) = O_{k}(1)e^{|t|C}⟨E⟩^{(k−1)/2}.  (3.5)

It follows that

U(t) ∈ C^{m}(R_{t}; L(D^{j}, D^{j+ \frac{1−m}{2}} )),  (3.6)

where the mth derivative is merely strongly continuous in t with values in L(D^{j}, D^{j− \frac{m−1}{2}} ), while the derivatives of lower order are continuous in the norm of L(D^{j}, D^{j+ \frac{1−m}{2}} ). Moreover,

∥∂_{t}^{k}U∥_{L(D^{j},D^{j+ \frac{1−k}{2}} )} ≤ C_{k,j}e^{C|t|}.  (3.7)

We take a break in the general discussion in order to recall some basic existence and uniqueness results for the abstract equation (3.3).

Proposition 3.1 If v ∈ C^{0}(R; D^{j− \frac{1}{2}} ), f ∈ D^{j}, g ∈ D^{j− \frac{1}{2}} , then

u(t) := U′(t)f + U(t)g + ∫_{0}^{t} U(t − s)v(s)ds

belongs to C^{0}(R; D^{j}) ∩ C^{1}(R; D^{j− \frac{1}{2}} ) ∩ C^{2}(R; D^{j−1}) and solves the Cauchy problem,

(∂_{t}^{2} + P)u = v, u(0) = f, u′(0) = g.  (3.8)

Proof. Notice that U(0) = 0, U′(0) = 1, so u_{1} := U′(t)f + U(t)g solves (3.8) with v = 0. Let

u_{2}(t) = ∫_{0}^{t} U(t − s)v(s).

Then u_{2}(0) = 0,

u′_{2}(t) = U(0)v(t) + ∫_{0}^{t} U′(t − s)v(s)ds = ∫_{0}^{t} U′(t − s)v(s)ds,

so u′_{2}(0) = 0. Since U′(0) = 1, we get

∂_{t}^{2}u_{2}(t) = v(t) + ∫_{0}^{t} U′′(t − s)v(s)ds,

and hence

∂_{t}^{2}u_{2}(t) + P u_{2}(t) = v(t) + ∫_{0}^{t}(∂_{t}^{2} + P)(U(t − s)v(s))ds = v(t),

since (∂_{t}^{2} + P)(U(t − s)v(s)) = 0. Further it is easy to see that u_{2}(t) belongs to the required space. #

Next look at the uniqueness of solutions of (3.8):

Proposition 3.2 Let I be an interval containing 0, and let u ∈ C^{2}(I; D^{k}) be a solution of (∂_{t}^{2} + P)u = 0. Then

u(t) = U′(t)u(0) + U(t)u′(0).  (3.9)

Proof. Let t ∈ I, and put v(s) = U(s − t)ϕ, ϕ ∈ D^{N} for some sufficiently large N. Then for s between 0 and t:

\frac{d}{ds}[(u′(s)|v(s)) − (u(s)|v′(s))] = (u′′(s)|v(s)) − (u(s)|v′′(s))
= −(P u(s)|v(s)) + (u(s)|P v(s)) = 0.

Hence

(u′(t)|v(t)) − (u(t)|v′(t)) = (u′(0)|v(0)) − (u(0)|v′(0)),

i.e.

−(u(t)|ϕ) = (u′(0)|U(−t)ϕ) − (u(0)|U′(−t)ϕ).

U(t) is odd in t, so U(−t) = −U(t), U′(−t) = U′(t). Hence

(u(t)|ϕ) = (u′(t)|U(t)ϕ) + (u(0)|U′(t)ϕ).

Use also that U(t) and U′(t) are symmetric:

(u(t)|ϕ) = (U(t)u′(0) + U′(t)u(0)|ϕ).

Since D^{N} is dense in D^{−k}, the space dual to D^{k}, we get (3.9). #

We now return to the main discussion. Outside \overline{B(0, R_{0})}, (3.3) is just the standard wave equation, for which Huygens’ principle applies and says that supports of solutions cannot propagate with speed > 1: If u ∈ C^{2}(R; D′(R^{n})) and

(−Δ + ∂_{t}^{2})u = 0, u(0) = u_{0}, ∂_{t}u(0) = u_{1},  (3.10)

then

supp u ⊂ {(t, x); dist_{R^{n}}(x, supp u_{0} ∪ supp u_{1}) ≤ |t|}.  (3.11)

In the bb setting, we define the support of u ∈ H_{loc} to be supp u_{|}_{R^{n}\B(0,R_{0}} if u vanishes near \overline{B(0, R_{0})} and \overline{B(0, R_{0})} ∪ supp u_{|}_{R^{n}\B(0,R_{0})} otherwise. Notice that supp u is closed.

Let dist_{bb}(x, y) = min(|x−y|, dist (x, B(0, R_{0}))+dist (y, B(0, R_{0}))). Then for (3.3) we have

supp u ⊂ {(t, x); dist_{bb}(x, supp g) ≤ |t|}.  (3.12)

We now introduce an abstract non-trapping condition, which says that singularities in compactly supported initial data are washed away from any bounded region after a suitable time:

For all a > R_{0}, ∃T_{a} > 0, such that t ↦ χU(t)χ  (3.13)
belongs to C^{∞}(]T_{a}, T_{a} + 1[; L(D^{0}; D^{N}))
for every N ≥ 0 and every χ ∈ C_{0}^{∞}(B(0, a)) with 1_{B(0,R_{0})} ≺ χ.

In practice, the verification of this non-trapping condition is important and depends on some result on the propagation of singularities for solutions of the wave equation. The condition (3.13) is known to hold in the following two cases: 1) P = −Δ + V, V ∈ C_{0}^{∞}(R^{n}) and 2) −P is the exterior Dirichlet or Neumann Laplacian on R^{n} \ O where O ⊂⊂ R^{n} is open with smooth boundary and non-trapping in the geometric sense that no maximal optical ray can be contained in a bounded set. For the second example we refer to [56], while the first example follows from the general result of Hörmander on progation of singularities for solutions of equations of principal type. See [42].

Let a > R_{0} + 1 and let 1\overline{B(0,R_{0}} ≺ χ_{a−1} ∈ C_{0}^{∞}(B(0, a − 1)). We will work for t ≥ 0 in the following. Let ψ_{a}(t) ∈ C_{0}^{∞}(] − ∞, T_{a} + 1[; [0, 1]) be equal to 1 near ] − ∞, T_{a}]. Then

(P + ∂_{t}^{2})(ψ_{a}(t)U(t)χ_{a−1}) =  (3.14)
(∂_{t}^{2}(ψ_{a}(t)) + 2∂_{t}(ψ_{a}(t))∂_{t})U(t)χ_{a−1} =: V_{a}(t).

In view of (3.13), we see that if 1\overline{B(0,R_{0})} ≺ χ ∈ C_{0}^{∞}(B(0, a)), then

χV_{a}(t) ∈ C_{0}^{∞}(]T_{a}, T_{a} + 1[; L(H, D^{N})), ∀N ∈ N.  (3.15)

We will choose χ ≺ χ_{a−1}.

The ”exterior part” (1 − χ)V_{a}(t)g will not be smooth in general, and we will add a correction for it, by solving the free wave equation. Let U_{0}(t) be the free wave ”group” defined as in (2.7) (with P replaced by −Δ and with H replaced by L^{2}(R^{n})). As explained in chapter 2, we have the corresponding forward fundamental solution E_{0} given by

E_{0} = H(t)U_{0}(t),  (3.16)

so that (∂_{t}^{2} − Δ)E_{0} = δ_{R^{n+1}}, and

supp E_{0} = \{ \begin{matrix}\{(t, x) ∈ R × R^{n}; |x| = t\}, for n odd ≥ 3, \\ \{(t, x) ∈ R × R^{n}; |x| ≤ t\}, otherwise. \end{matrix}  (3.17)

Notice that V_{a}g vanishes outside some bounded set, by Huygens’ principle. Let χ ∈ C_{0}^{∞}(B(0, a−1)) be equal to 1 near \overline{B(0, R_{0})}. (We will take χ ≺ χ_{a−1}.) When n ≥ 3 is odd, the corrected truncated solution of the wave equation (associated to P) is then by definition:

Ũ_{a}(t) = ψ_{a}(t)U(t)χ_{a−1} − (1 − χ̃)(E_{0}∗)(1 − χ)V_{a}(t),  (3.18)

where E_{0}∗ is the operator of convolution with E_{0}, and χ̃ ∈ C_{0}^{∞} with 1\overline{B(0,R_{0})} ≺ χ̃ ≺ χ. We get

(P + ∂_{t}^{2})Ũ_{a}(t)g = χV_{a}(t)g + [−Δ, χ̃](E_{0}∗)(1 − χ)V_{a}(t)g.  (3.19)

The second term has uniformly compact support by the strong Huygens principle. Let us consider also

Û_{a}(t)g := ψ_{a}(t)U(t)χ_{a−1}g − (E_{0}∗)(1 − χ)V_{a}(t)g  (3.20)

as a distribution in (R × (R^{n} \ \overline{B(0, R_{0})})) ∪ (]T_{a} + 1, ∞[×R^{n}), so that

(∂_{t}^{2} − Δ)Û_{a}(t)g ∈ C^{∞}(R × (R^{n} \ B(0, R_{0}))) ∪ (]T_{a} + 1, ∞[×R^{n})) (3.21)

Now we shall use a fact about propagation of singularities for solutions of the wave equation. Let Ω ⊂ R×R^{n} be open, u ∈ D′(Ω), (∂_{t}^{2}−Δ)u ∈ C^{∞}(Ω):

If (t_{0}, x_{0}) ∈ sing supp (u) then there exists a maximally extended light ray I ∋ t ↦ (t_{0} + t, x_{0} + tω) in Ω, where I is an open interval containing 0, ω ∈ S^{n−1}, which passes through (t_{0}, x_{0}) and is contained in sing supp u.

Let (t_{0}, x_{0}) ∈ sing supp (Û_{a}(t)g), with R_{0} < |x_{0}| < a − 1, t_{0} > T_{a}. For t > T_{a}, r_{0} < |x| < a, the term ψ_{a}(t)U(t)χ_{a−1}g is in C^{∞}, so (t_{0}, x_{0}) must belong to sing supp E_{0}∗(1−χ)V_{a}(t)g ⊂ {(t, x); (a−|x|)_{+} ≤ t−T_{a}}. We deduce that t_{0} > T_{a} + 1. Also from the fact above, we see that there is a maximally extended ray in (R × (R^{n} \ B(0, R_{0}))) ∪ (]T_{a} + 1, ∞[×R^{n}) passing through (t_{0}, x_{0}), contained in sing supp (Û_{a}(t)g), which is well defined at t = T_{a} + 1 and with a corresponding point (T_{a} + 1, x(T_{a} + 1)) with |x(T_{a} + 1)| ≥ a − 1. But then it is clear that the ray can be further extended backwards for all times t < T_{a} + 1 and the corresponding x(t) stays outside B(0, a − 1). This ray must be in sing supp Û_{a}(t)g and we reach a contradiction for |t| close to 0, since we are then outside the support of Û_{a}(t)g.

In conclusion: Û_{a}(t)g and (E_{0}∗)(1 − χ)V_{a}(t)g are C^{∞} for t > T_{a}, R_{0} < |x| < a − 1. This implies that the last term in (3.19) is C^{∞} for |x| < a − 1. Moreover, by the strong Huygens principle, it has compact support. So

(P + ∂_{t}^{2})Ũ_{a}(t)g = W_{a}(t)g,  (3.22)

where W_{a}(t)g ∈ C_{0}^{∞}(]0, ∞[; D_{comp}^{N}) has its support in some fixed compact set contained in ]T_{a}, T̂_{a}[×supp χ. (When n is even, we loose the compact support property in t.)

By the closed graph theorem,

W_{a} ∈ C_{0}^{∞}(]0, ∞[; L(H, D_{comp}^{N})).  (3.23)

We also have the Cauchy data,

Ũ_{a}(t)g = 0, (∂_{t}Ũ_{a})(0)g = χ_{a−1}g.  (3.24)

Finally Ũ_{a}(t) has the same general regularity properties as U_{a}:

Ũ_{a}(t) ∈ C^{m}(R_{t}; L(D^{j}, D^{j+ \frac{1−m}{2}} )), m ≥ 0, j ≥ 0.  (3.25)

(From (3.22), (3.24), we deduce that Ũ_{a} − Uχ_{a−1} ∈ C^{∞}(R; L(H, D^{j})), for all j ≥ 0.)

When Im λ ≥ C ≫ 0, we express R(λ) = (P − λ^{2})^{−1} by the integral formula (cf. (3.7)):

R(λ) = ∫_{0}^{∞} e^{itλ}U(t)dt.  (3.26)

In fact,

−λ^{2}R(λ) = ∫_{0}^{∞} ∂_{t}^{2}(e^{itλ})U(t)dt  (3.27)
= − ∫_{0}^{∞} ∂_{t}(e^{itλ})∂_{t}U(t)dt = ∂_{t}U(0) + ∫_{0}^{∞} e^{itλ}∂_{t}^{2}U(t)dt
= 1 + ∫_{0}^{∞} e^{itλ}∂_{t}^{2}U(t)dt,

and it follows that

(P − λ^{2}) ∫_{0}^{∞} e^{itλ}U(t)dt = 1.

Put

R_{a}(λ) = ∫_{0}^{∞} e^{itλ}Ũ_{a}(t)dt : H → D_{loc},  (3.28)

and notice that this operator is well defined and holomorphic for λ ∈ C, since by the strong Huygens principle,

(Ũ_{a}(t)g)(x) = 0, for t ≥ T̃_{a} + |x|.  (3.29)

The same calculation gives

(P − λ^{2})R_{a}(λ) = χ_{a−1} + ∫_{0}^{∞} e^{itλ}W_{a}(t)dt.  (3.30)

Since χ̃ ≺ χ ≺ χ_{a−1}, the x-space projection of supp W_{a}(t) is contained in the region where χ_{a−1} = 1. Writing

S_{a}(λ) = ∫_{0}^{∞} e^{itλ}W_{a}(t)gdt,  (3.31)

we get

(P − λ^{2})R_{a}(λ) = χ_{a−1}(1 + S_{a}(λ)).  (3.32)

Lemma 3.3 For every N > 0, 1 + S_{a}(λ) : H → H, H_{comp} → H_{comp} is invertible with inverse O(1) in L(H, H), for

Im λ ≥ −N log |λ|, |λ| ≥ C(N).

Proof. We use the compact support property and (3.23) for W_{a}, and integrate by parts in (3.31):

S_{a}(λ) = \frac{1}{(iλ)^{M}} ∫_{0}^{∞} e^{itλ}(−∂_{t})^{M}W_{a}(t)dt.

The H → H norm of this operator is ≤ C_{M}|λ|^{−M}e^{_{T_{a}}(Im λ)_{−}}, with the standard notation b = b_{+} − b_{−}, b_{±} = max(±b, 0). If Im λ ≥ −N log |λ|, we get

∥S_{a}(λ)∥_{L(H,H)} ≤ C_{M}|λ|^{_{T_{a}}N−M},

which is ≤ \frac{1}{2} if we first take M > T̂_{a}N, and then |λ| large enough. Notice that (1 + S_{a}(λ))^{−1} = 1 − S_{a}(λ)(1 + S_{a}(λ))^{−1} maps H_{comp} → H_{comp}. #

When Im λ ≥ C ≫ 0, we get from (3.32) the relation

R(λ)χ_{a−1} = R_{a}(λ)(1 + S_{a}(λ))^{−1},  (3.33)

for R(λ) = (P − λ^{2})^{−1}, and we can use this relation to get a meromorphic extension of R(λ)χ_{a−1} to C, (since S_{a}(λ) is compact and holomorphic in λ). If (as before), we let R(λ) denote also the meromorphic extension of R(λ) : H_{comp} → H_{loc}, then (3.33) extends to λ ∈ C, and we conclude that R(λ)χ_{a−1} has at most finitely many poles in Im λ ≥ −N log_{+} |λ|, for every N ≥ 0. Applying Proposition 2.3, we then get the following result (which remains true for n even).

Proposition 3.4 In addition to the general bb assumptions of chapter 2, we assume that P ≥ −C and that the non-trapping assumption (3.13) holds. We also assume that n ≥ 3 is odd. Let R(λ) be the meromorphic extension of (P − λ^{2})^{−1} : H_{comp} → D_{loc}. Then for every N ≥ 0, R(λ) has at most finitely many poles in Im λ ≥ −N log_{+} |λ|.

We next estimate the truncated extension of R(λ) and start with R_{a}. If χ ∈ C_{0}^{∞}(R^{n}) is equal to 1 near \overline{B(0, R_{0})}, then (by the strong Huygens principle) χŨ_{a}(t) vanishes for t ≥ T_{χ} for some sufficiently large T_{χ} > 0. Hence

χR_{a}(λ) = ∫_{0}^{T_{χ}} e^{itλ}χŨ_{a}(t)dt,  (3.34)

and using (3.25), we get after an integration by parts:

∥χR_{a}(λ)∥_{L(D^{j},D^{j})} ≤ \frac{C_{j}}{|λ|}e^{T_{χ}(Im λ)_{−}}, j ≥ 0.  (3.35)

Combining this with (3.33) and Lemma 3.3, we get

∥χR(λ)χ_{a−1}∥_{L(H,H)} ≤ \frac{C}{|λ|}e^{T_{χ}(Im λ)_{−}},  (3.36)

for |λ| ≥ C(N), Im λ ≥ −N log_{+} |λ|. Here we will need to improve the exponent −1 in |λ|^{−1} by replacing ∥ · ∥_{L(H,H)} by ∥ · ∥_{L(D^{θ},H)}

As a preparation for the promised asymptotic expansion of the wave group for t → +∞, we recall that

R(λ) = ∫_{0}^{∞} e^{itλ}U(t)dt, Im λ ≫ 0,

and we shall establish a corresponding ”Fourier inversion” formula, which could be obtained by selfadjoint functional calculus, but which we prefer to derive by direct arguments. For this, we work in a region Im λ ≥ C ≫ 0 and start by getting some further estimates on R(λ).

First, by (3.7), (3.26), we get

∥R(λ)∥_{L(D^{j},D^{j+ \frac{1}{2}} )} ≤ O(1),  (3.37)

and after integration by parts in (3.26),

∥R(λ)∥_{L(D^{j},D^{j})} ≤ \frac{O(1)}{⟨λ⟩}.  (3.38)

From the first of the two equivalent identities

P R(λ) = 1 + λ^{2}R(λ), R(λ) = −\frac{1}{λ^{2}} + \frac{1}{λ^{2}} P R(λ),  (3.39)

and (3.38), we get

∥R(λ)∥_{L(D^{j},D^{j+1})} ≤ O(1)⟨λ⟩.  (3.40)

Use (3.39), (3.37) to get

∥R(λ)∥_{L(D^{j},D^{j−1/2})} ≤ \frac{O(1)}{⟨λ⟩^{2}}.  (3.41)

The last two estimates are the most extreme ones, and (3.37), (3.38) can be obtained from these two by interpolation (cf (3.2)) and we get more generally

∥R(λ)∥_{L(D^{j},D^{j−α})} ≤ \frac{O(1)}{⟨λ⟩^{1+2α}} , −1 ≤ α ≤ \frac{1}{2}.  (3.42)

We claim that U(t) = Û(t), where

Û(t) = \frac{1}{2π} ∫_{Im =C} e^{−itλ}R(λ)dλ, t ≥ 0.  (3.43)

The integral converges in L(D^{j}, D^{j−ϵ}) for every ϵ > 0 and becomes a continuous function of t with values in that space. Using (3.39), we get

Û(t)g = \frac{1}{2π} ∫_{Im λ=C} e^{−itλ}(−\frac{1}{λ^{2}} + \frac{R(λ)}{λ^{2}}P)gdλ
= \frac{1}{2π} ∫_{γ} e^{−itλ} \frac{1}{λ^{2}} dλ g + \frac{1}{2π} ∫_{Im λ=C} e^{−itλ} \frac{R(λ)}{λ^{2}}P gdλ
= tg + \frac{1}{2π} ∫_{Im λ=C} e^{−itλ} \frac{R(λ)}{λ^{2}}P gdλ,

where γ is a small positively oriented circle around 0.

In the last integral, we gained a sufficiently large negative power of λ, to be able to apply ∂^{2} and get

∂_{t}^{2}Û(t)g = −\frac{1}{2π} ∫_{Im λ=C} e^{−itλ}R(λ)P gdλ = −P Û(t)g,

i.e.

(P + ∂_{t}^{2})Û(t)g = 0.  (3.44)

Further,

Û(0)g = \frac{1}{2π} ∫_{Im λ=C} \frac{R(λ)}{λ^{2}}P gdλ = 0,

by contour deformation upwards, and

∂_{t}Û(0)g = g − \frac{i}{2π} ∫_{Im λ=C} \frac{R(λ)}{λ}P gdλ = g,

by the same contour deformation.

We have then verified that Û(t) solves the same Cauchy problem as U, so we have U = Û:

U(t) = \frac{1}{2π} ∫_{Im λ=C} e^{−itλ}R(λ)dλ, t ≥ 0,  (3.45)

where the integral converges in L(D^{j}, D^{j−ϵ}) for every ϵ > 0.

We want to multiply this to the left by χ and to the right by χ_{a−1}, and then make contour deformation, and use an estimate like (3.36), but with a little faster decay than |λ|^{−1}, so we start by obtaining such improved decay. For g ∈ D we write

χR(λ)χ_{a−1}g = χ(−\frac{1}{λ^{2}} + \frac{R(λ)}{λ^{2}} χ̃_{a−1}P)χ_{a−1}g,

where χ_{a−1} ≺ χ̃_{a−1} ∈ C_{0}^{∞}(B(0, a−1)). Using (3.36) (with χ_{a−1} there replaced by χ̃_{a−1}) we get for the same λ as there,

∥χR(λ)χ_{a−1}∥_{L(D,H)} ≤ \frac{C}{|λ|^{2}} e^{T_{χ}(Im λ)_{−}}.  (3.46)

(A more optimal estimate would probably be to have L(D^{1/2}, H) instead of L(D, H).) Interpolating between (3.36) and (3.46), we get

∥χR(λ)χ_{a−1}∥_{L(D^{θ},H)} ≤ \frac{C}{|λ|^{1+θ}} e^{T_{χ}(Im λ)_{−}},  (3.47)

for λ as in (3.36) and 0 ≤ θ ≤ 1.

With this in mind, we write

χU(t)χ_{a−1}g = \frac{1}{2π} ∫_{Im λ=C} e^{−itλ}χR(λ)χ_{a−1}dλ  (3.48)

= \frac{1}{2π} ∫_{Im λ=−A} e^{−itλ}χR(λ)χ_{a−1}dλ + ∑_{Im λ>−A} residues,

where we choose A > 0 such that no resonances has the imaginary part −A. The L(D^{θ}, H)-norm of the integral in the last expression in (3.48) can be estimated by

O_{A}(1)e^{−tA+T_{χ}A} = O_{A}(1)e^{−At}.

Let λ_{0} ∈ C, be a pole of the meromorphic extension of (P − λ^{2})^{−1}, so that in the case λ_{0} ≠ 0 (cf (2.48)) we have near λ_{0}:

−R(λ) = ∑_{1}^{ω(λ_{0})} \frac{Π_{−j}}{(λ^{2} − λ_{0}^{2})^{j}} + Hol(λ)  (3.49)

In the case Im λ_{0} ≤ 0 we discussed some of the structure of the ”spectral projections” Π_{−j}. When Im λ_{0} > 0 we know that λ_{0} is purely imaginary and that λ_{0}^{2} is a negative eigenvalue of P. In this case Π_{−1} is the ordinary orthogonal spectral projection, while Π_{−j} = 0 for j ≥ 2. In the case λ_{0} = 0 things are a little more complicated and we make no further comments in that case. For λ_{0} ≠ 0 the corresponding contribution to the sum of residues in (3.48) is

∑_{1}^{ω(λ_{0})} \frac{1}{2π} ∫_{γ_{0}} e^{−itλ} \frac{1}{(λ^{2} − λ_{0}^{2})^{j}} dλ χΠ_{−j}χ_{a−1},  (3.50)

where γ_{0} is a small positively oriented circle around λ_{0}. The integral is of the form e^{−itλ_{0}} times a polynomial of degree at most j − 1 in t. For j = 1, we get

\frac{1}{2π} 2πi\frac{e^{−itλ_{0}}}{2λ_{0}} χΠ_{−1}χ_{a−1} = \frac{ie^{−itλ_{0}}}{2λ_{0}} χΠ_{−1}χ_{a−1}.  (3.51)

We have proved the following result essentially due to Lax and Phillips with a more straight forward proof due to Vainberg [91], extended to the black box framework by Tang-Zworski [90], that we have modified a little to decrease the number of cut-offs:

Theorem 3.5 Let n ≥ 3 be odd. Let P be a black box operator as in chapter 2 and assume (3.1), (3.13). Then for every A > 0 such that −A is not the imaginary part of any resonance, and for every χ ∈ C_{0}^{∞}(R^{n}) equal to a constant near \overline{B(0, R_{0})}, we have

χU(t)χ = ∑_{λ∈Res (P)∪{iμ;μ>0, −μ^{2}∈σ(P))}} e^{−itλ}χp_{λ,P} (t)χ + r_{χ,A}(t),

where p_{λ,P} (t) is a polynomial in t with values in the operators H_{comp} → D_{loc}. When λ_{0} is a simple pole of the resolvent and λ_{0} ≠ 0, we have p_{λ,P} (t) = \frac{i}{2λ}Π_{−1,λ}, where Π_{−1,λ} is the corresponding spectral projection. The remainder term satisfies ∥r_{χ,A}(t)∥_{L(D^{θ},H)} ≤ C_{χ,A,θ}e^{−tA}, for every θ > 0.

According to Tang-Zworski, Burq, there is a trick of C. Morawetz, which allows us to replace D^{θ} by H in the above theorem.

4 Absence of resonances exponentially close to R.

In this chapter we shall descibe a recent result of N. Burq [15] which says that under quite general assumptions there are no resonances exponentially close to the real axis. We start by formulating the precise theorem. Let O ⊂⊂ R^{n} be an open set with a smooth (C^{∞}) boundary ∂O and assume that R^{n} \ O is connected. Let

P = ∑_{|α|≤2} a_{α}(x)D^{α}  (4.1)

be an elliptic formally selfadjoint operator with smooth coefficients on R^{n}\O. We assume that

P = −Δ for |x| > R_{0} − 1,  (4.2)

for some sufficiently large R_{0} > 0, that we choose large enough so that \overline{O} ⊂ B(0, R_{0} − 1).

We explain the main ideas of the proof of the following result of N. Burq [15].

Theorem 4.1 Let P also denote the Dirichlet realization of P in L^{2}(R^{n}\O). Then there exists a constant C > 0, such that P has no resonances in the set of λ ∈ C such that

|λ| ≥ C, Im λ ≥ −e^{−C|Re λ|}.  (4.3)

Actually, Burq also establishes this result for the Neumann realization of P and even in the more general case when P is realized with Dirichlet conditions on one connected part of the boundary and with Neumann conditions on the other part. As a corollary, Burq shows that the local energy of solutions to the wave equation with compactly supported initial data decays at least as fast as (log(1 + t))^{−k}, t → +∞, where k ≥ 0 depends on the smoothness of the Cauchy data at t = 0. Here smoothness of the data is measured in terms of being in the domain of a certain power of the generator of the wave group, when the latter is considered as a matrix solution to a first order system. Earlier results in this direction were obtained by E. Harrell [36] and C. Fernandez and R. Lavine [24]. After the work [15] there have been extensions by G. Vodev [97, 98], and by Burq [16]. Vodev observed some simplifications in Burq’s original proof, and we shall use one of them in our presentation, as well as a new estimate (that will not be explained in detail here) that is planned to appear in a joint work with Burq. This estimate permits to give a treatment near infinity which is more natural and closer to the intuition.

The first step in the proof is to notice that it suffices to show that for real λ with |λ| ≥ 1, we have

∥R(λ)v∥_{H^{1}(B(0,R_{0})\O)} ≤ C_{0}e^{C_{0}|λ|}∥v∥_{L^{2}},  (4.4)

for v ∈ L_{comp}^{2}(B(0, R_{0}) \ O) for some C_{0} > 0. In fact, assuming (4.4), we shall extend R(μ) holomorphically to μ in an exponentially small disc centered at λ as an operator L_{comp}^{2}(R^{n} \ O) → (H^{2} ∩ H_{0}^{1})_{loc}(R^{n} \ O). Let 1_{B(0,R_{0}−1)} ≺ χ_{0} ≺ χ_{1} ≺ χ_{2} ≺ 1_{B(0,R_{0})}, with χ_{j} ∈ C_{0}^{∞}. As an approximation for R(μ)χ_{1}, we take

A(μ) := χ_{2}R(λ)χ_{1} − R_{0}(μ)[P_{0}, χ_{2}]R(λ)χ_{1}.

Then

(P − μ^{2})A(μ) =  (4.5)
χ_{1} + [P_{0}, χ_{2}]R(λ)χ_{1} + χ_{2}(λ^{2} − μ^{2})R(λ)χ_{1} − [P_{0}, χ_{2}]R(λ)χ_{1}
= χ_{1} + χ_{2}(λ^{2} − μ^{2})R(λ)χ_{1}.

For the exterior part of the approximate resolvent, we first compute

(P − μ^{2})(1 − χ_{0})R_{0}(μ)(1 − χ_{1}) = 1 − χ_{1} − [P_{0}, χ_{0}]R_{0}(μ)(1 − χ_{1}).

Correcting for the last term, we are led to the approximation of R(μ)(1−χ_{1}) :

B(μ) := (1 − χ_{0})R_{0}(μ)(1 − χ_{1}) + A(μ)[P_{0}, χ_{0}]R_{0}(μ)(1 − χ_{1}).

Then

(P − μ^{2})B(μ) = 1 − χ_{1} − [P_{0}, χ_{0}]R_{0}(μ)(1 − χ_{1})
+(χ_{1} + χ_{2}(λ^{2} − μ^{2})R(λ)χ_{1})[P_{0}, χ_{0}]R_{0}(μ)(1 − χ_{1})
= (1 − χ_{1}) + χ_{2}(λ^{2} − μ^{2})R(λ)[P_{0}, χ_{0}]R_{0}(μ)(1 − χ_{1}).

Put R̃(μ) = A(μ) + B(μ). Then

(P − μ^{2})R̃(μ) = 1 + K,  (4.6)
K = χ_{2}(λ^{2} − μ^{2})R(λ)(χ_{1} + [P_{0}, χ_{0}]R_{0}(μ)(1 − χ_{1})).  (4.7)

Using the Fourier transform and complex deformation of integration contours as in section 2.1, we see that

[P_{0}, χ_{0}]R_{0}(μ) : L_{comp}^{2}(B(0, R_{0}) \ O) → L_{comp}^{2}(B(0, R_{0}) \ O)

is of norm ≤ Ce^{C|μ|}. It follows that

K : L_{comp}^{2}(B(0, R_{0}) \ O) → L_{comp}^{2}(B(0, R_{0}) \ O)

and has operator norm O(1)|λ|e^{C_{0}|λ|}|λ−μ|, so if |λ−μ| ≤ e^{−2C_{0}|λ|} and |λ| ≫ 1, 1 + K has a bounded inverse and we get the holomorphic extension

R(μ) = R̃(μ)(1 + K)^{−1} : L_{comp}^{2}(B(0, R_{0}) \ O) → (H^{2} ∩ H_{0}^{1})_{loc}(B(0, R_{0}) \ O). (4.8)

Proposition 2.3 then shows that R(μ) extends holomorphically also as an operator on L_{comp}^{2}(R^{n} \ O).

One of the main steps of the proof is to establish weighted L^{2} estimates in a bounded domain for (P − λ^{2}) = λ^{2}(h^{2}P − 1), with h = 1/|λ|. These estimates are so called Carleman estimes, developed with enormous success by Hörmander and others in different branches of partial differential equations, including uniqueness problems and Cauchy-Riemann type equations. In the context of spectral problems, they have been used by G. Lebeau and L. Robbiano [52, 53]. Much works in a more general frame work, so to start with, we take a more general semi-classical operator.

4.1 Local Carleman estimates away from the boundary.

Let Ω ⊂ R^{n} be some open set. Let P be a semiclassical differential operator of order 2, formally self-adjoint and of the form:

P = ∑_{|α|≤2} a_{α}(x; h)(hD_{x})^{α}, a_{α}(x; h) ∈ C^{∞}(Ω).  (4.9)

Assume that

a_{α}(x; h) = a_{α}^{0}(x) + O(h) in C^{∞}(Ω),  (4.10)

and that a_{α} is independent of h for |α| = 2. The semiclassical principal symbol of P is then by definition:

p(x, ξ) = ∑_{|α|≤2} a_{α}^{0}(x)ξ^{α} = a(x, ξ) + ∑_{|α|≤1} a_{α}^{0}(x)ξ^{α}.  (4.11)

Here a(x, ξ) = ∑_{|α|=2}(...) is equal to the ordinary principal symbol of the operator h^{−2}P. We assume that P is elliptic in the classical sense:

a(x, ξ) ≥ \frac{1}{C} |ξ|^{2}.  (4.12)

Let ϕ ∈ C^{∞}(Ω; R) and consider

P_{ϕ} = e^{ϕ(x)/h}P e^{−ϕ(x)/h}.  (4.13)

This operator is also of the form (4.9) with the same ordinary principal symbol h^{2}a(x, ξ) and with the new semiclassical principal symbol

p_{ϕ}(x, ξ) = p(x, ξ + iϕ′(x)).  (4.14)

Assume that we have

p_{ϕ}(x, ξ) = 0 ⇒ \frac{1}{i}\{p_{ϕ}, \overline{p_{ϕ}}\} < 0,  (4.15)

where

\{f, g\} = ∑_{j=1}^{n}(\frac{∂f}{∂ξ_{j}} (x, ξ)\frac{∂g}{∂x_{j}} (x, ξ) − \frac{∂f}{∂x_{j}} (x, ξ)\frac{∂g}{∂ξ_{j}} (x, ξ)),

denotes the Poisson bracket of two differentiable functions f, g. Notice that \frac{1}{i}\{f, \overline{f}\} = −2\{Re f, Im f\} is a real-valued function.

The basic Carleman estimate is then given by:

Lemma 4.2 Let W ⊂⊂ Ω be open. Then there exist constants C_{0}, h_{0} > 0 such that for all 0 < h ≤ h_{0} , u ∈ C_{0}^{∞}(W):

h^{1/2}∥u∥_{H^{2}} ≤ C_{0}∥P_{ϕ}u∥.  (4.16)

Here we let H^{s} denote the semi-classical Sobolev space equipped with the hdependent norm ∥u∥_{H^{s}} = ∥⟨hD⟩^{s}u∥.

We will only give an outline of the proof, mainly for readers who are familiar with pseudodifferential operator machinery especially in the semiclassical setting. However, we point out that some basic ideas are older than the theory of pseudodifferential operators. Start by looking at the trivial identity:

\frac{1}{h}∥P_{ϕ}u∥^{2} ≥ \frac{1}{h}∥P_{ϕ}u∥^{2} − \frac{1}{h}∥P_{ϕ}^{∗}u∥^{2}  (4.17)

= \frac{1}{h}((P_{ϕ}^{∗}P_{ϕ} − P_{ϕ}P_{ϕ}^{∗})u|u) = (\frac{1}{h}[P_{ϕ}^{∗}, P_{ϕ}]u|u).

Here the commutator has the form

\frac{1}{h}[P_{ϕ}^{∗}, P_{ϕ}] = ∑_{|α|≤3} b_{α}(x; h)(hD_{x})^{α},

with

b_{α}(x; h) = b_{α}^{0}(x) + O(h) in C^{∞}(Ω),

and the associated semi-classical principal symbol is

\frac{1}{i}\{\overline{p_{ϕ}}, p_{ϕ}\}

which is > 0 on p_{ϕ}^{−1}(0). Since |p_{ϕ}(x, ξ)| ∼ ⟨ξ⟩^{2} for large ξ, we can find constants b, C̃ > 0, such that

\frac{1}{i}\{\overline{p_{ϕ}}, p_{ϕ}\} + b|p_{ϕ}(x, ξ)|^{2} ≥ \frac{1}{C̃} ⟨ξ⟩^{4} on W̃ × R^{n},

where we choose W̃ ⊂⊂ Ω open with W ⊂⊂ W̃.

Using h-pseudodifferential calculus and/or a suitable version of Gårding’s inequility (see for instance [21]), we deduce that

(\frac{1}{h}[P_{ϕ}^{∗}, P_{ϕ}]u|u) + b∥P_{ϕ}u∥^{2} ≥ \frac{1}{2C̃} ∥u∥_{H^{2}}^{2}, u ∈ C_{0}^{∞}(W),

We get

∥u∥_{H^{2}}^{2} ≤ 2C̃((\frac{1}{h}[P_{ϕ}^{∗}, P_{ϕ}]u|u) + b∥P_{ϕ}u∥^{2}).  (4.18)

Combining this with (4.17), we get (4.16). #

Actually, we shall not use the full strength of (4.16) but only the weaker estimate

h^{1/2}∥u∥_{H^{1}} ≤ O_{W} (1)∥P_{ϕ}u∥, u ∈ C_{0}^{∞}(W).  (4.19)

Taking squares and recalling the definition of P_{ϕ}, we get

h ∫ (e^{2ϕ/h}|u|^{2} + |h∇_{x}(e^{ϕ/h}u)|^{2})dx ≤ O_{W} (1) ∫ e^{2ϕ/h}|P u|^{2}dx.  (4.20)

This can also be written:

h ∫ e^{2ϕ/h}(|u|^{2} + |h∇_{x}u|^{2})dx ≤ O_{W} (1) ∫ e^{2ϕ/h}|P u|^{2}dx.  (4.21)

4.2 Local Carleman estimates at the boundary

Now let Ω ⊂ R^{n} be closed, bounded and have a smooth boundary ∂Ω, Let x_{0} ∈ ∂Ω and assume that P, ϕ are as above in a neighborhood W (in R^{n}) of x_{0}. We want to have (4.21) for u ∈ C_{0}^{∞}(Ω ∩ W) with u_{|}_{∂Ω} = 0. Using the ellipticity, it is quite classical that we can find local coordinates x_{1}, .., x_{n} centered at x_{0}, such that Ω is defined in W by x_{n} ≥ 0, and the (new) principal symbol becomes

a(x, ξ) = ξ_{n}^{2} + r(x, ξ′), ξ′ = (ξ_{1}, .., ξ_{n−1}).  (4.22)

So

P = (hD_{x_{n}})^{2} + R(x, hD_{x′}; h) + b(x; h)hD_{x_{n}}.  (4.23)

The Lebesgue measure becomes λ(x)dx for some smooth and strictly positive function λ(x). Define P_{ϕ} as above. It is now practical to organize the computations a little differently, so we write

P_{ϕ} = Q_{2} + iQ_{1}, Q_{2} = \frac{1}{2}(P_{ϕ} + P_{ϕ}^{∗}), Q_{1} = \frac{1}{2i}(P_{ϕ} − P_{ϕ}^{∗}).

Here the * indicates that the we take the formal adjoint in L^{2}(W; λ(x)dx_{1}..dx_{n}) and norms and scalar products will be in this space for a while. The subscript j in Q_{j} indicates that we have a differential operator of order j.

Then for all u ∈ C_{0}^{∞}(Ω ∩ W) with u_{|}_{∂Ω} = 0, we get

∥P_{ϕ}u∥^{2} = ∥Q_{2}u∥^{2} + ∥Q_{1}u∥^{2} + i[(Q_{1}u|Q_{2}u) − (Q_{2}u|Q_{1}u)].  (4.24)

If we could move the Q_{j} from the right to the left in each of the two scalar products, the bracket in (4.24) would become

(i[Q_{2}, Q_{1}]u|u) = \frac{1}{2}([P_{ϕ}^{∗}, P_{ϕ}]u|u),

and we can hope to estimate this from below as before. (The presence of the factor 1/2 is explained by the fact that ∥Q_{2}u∥^{2} + ∥Q_{1}u∥^{2} is not in general equal to ∥P_{ϕ}^{∗}u∥^{2}, but rather to \frac{1}{2}∥P_{ϕ}u∥^{2} + \frac{1}{2}∥P_{ϕ}^{∗}u∥^{2}.) However, we have to take into account the boundary terms, when carrying out the integrations by parts that pull over the Q_{j} to the left. Because of the Dirichlet condition, we get no boundary integral, when moving over Q_{1}, while Q_{2} will give rise to a boundary term from its (hD_{x_{n}})^{2} term. Notice that Q_{2} = (hD_{x_{n}})^{2} plus terms which contain hD_{x_{n}} at most to the power 1. Hence,

i(Q_{1}u|Q_{2}u) = i(Q_{2}Q_{1}u|u) + h ∫_{x_{n}=0}(Q_{1}u)\overline{hD_{x_{n}}u}λ(x′, 0)dx′.  (4.25)

Only the hD_{x_{n}} term in Q_{1} can give a non-vanishing contribution to the boundary integral (because of the Dirichlet condition) so we compute this term, using (4.23):

P_{ϕ} = (hD_{x_{n}} + i∂_{x_{n}}ϕ)^{2} + R_{ϕ}(x, hD_{x′}; h) + b(x; h)(hD_{x_{n}} + i∂_{x_{n}}ϕ)
= (hD_{x_{n}})^{2} − (∂_{x_{n}}ϕ)^{2} + i((∂_{x_{n}}ϕ) ∘ hD_{x_{n}} + (hD_{x_{n}}) ∘ (∂_{x_{n}}ϕ)) +
R_{ϕ}(x, hD_{x′}; h) + bhD_{x_{n}} + ib∂_{x_{n}}ϕ
= P − (∂_{x_{n}}ϕ)^{2} + (R_{ϕ} − R) +
i((∂_{x_{n}}ϕ) ∘ hD_{x_{n}} + (hD_{x_{n}}) ∘ (∂_{x_{n}}ϕ)) + ib∂_{x_{n}}ϕ.

Now use that (hD_{x_{n}})^{∗} = hD_{x_{n}} + hμ(x) for some smooth function μ (which depends on λ) and

P_{ϕ}^{∗} = P − (∂_{x_{n}}ϕ)^{2} + (R_{ϕ}^{∗} − R^{∗}) − i((hD_{x_{n}})^{∗} ∘ (∂_{x_{n}}ϕ) + (∂_{x_{n}}ϕ)(hD_{x_{n}})^{∗})
−ib∂_{x_{n}}ϕ,

to conclude that the hD_{x_{n}} term in P_{ϕ} − P_{ϕ}^{∗} is 4i(∂_{x_{n}}ϕ)hD_{x_{n}}, so

Q_{1} = \frac{1}{2i}(P_{ϕ} − P_{ϕ}^{∗}) = 2(∂_{x_{n}}ϕ) hD_{x_{n}} + a tangential operator,  (4.26)

where by a tangential operator we mean one which contains differentiations with respect to x_{1}, .., x_{n−1} but not with respect to x_{n}.

Now (4.25) becomes

i(Q_{1}u|Q_{2}u) = i(Q_{2}Q_{1}u|u) + 2h ∫_{x_{n}=0}(∂_{x_{n}}ϕ)|hD_{x_{n}}u|^{2}λ(x′, 0)dx′. (4.27)

Using this in (4.24), we get

∥P_{ϕ}u∥^{2} = ∥Q_{2}u∥^{2} + ∥Q_{1}u∥^{2} +  (4.28)
2h ∫_{x_{n}=0}(∂_{x_{n}}ϕ)|hD_{x_{n}}u|^{2}λ(x′, 0)dx′ + (i[Q_{2}, Q_{1}]u|u),

so we can hope to have (4.21) also in the boundary case, provided that

∂_{x_{n}}ϕ_{|}_{x_{n}=0} > 0.  (4.29)

A coordinate free formulation of the last condition is that

\frac{∂ϕ}{∂ν}_{|}_{∂Ω} > 0,  (4.30)

where ν denotes the interior unit normal of ∂Ω, for the metric given by the quadratic form on the tangent space which is dual to a(x, ξ).

The treatment of the last term in (4.28) becomes more technical because of the boundary (and requires also the use of tangential pseudodifferential operators and a division trick going back to a work of V. Ivrii). This was carried out by Lebeau–Robbiano [52, 53] also in the case of Neumann boundary conditions. We will not go into the technical details and simply state the result, as it was formulated by Burq [15]: Let P be as above, but of a more restricted form:

P = ∑_{|α|=2} a_{α}(x)(hD_{x})^{α} + h ∑_{|α|≤1} b_{α}(x; h)(hD_{x})^{α} − 1,  (4.31)

with b_{α}(x; h) = O(1) in C^{∞}(Ω). Then we have Proposition 4.3 Let P, ϕ, Ω be as above (including (4.30), (4.31)) and let W be a sufficiently small neighborhood of x_{0} ∈ ∂Ω. Then for h > 0 small enough, and all u ∈ C_{0}^{∞}(Ω ∩ W) with u_{|}_{∂Ω} = 0, we have (4.20), (4.21), with Ω ∩ W as the integration domain.

4.3 Gluing local Carleman estimates for a given ϕ.

Let O ⊂ R^{n} be closed with smooth boundary, and let W ⊂⊂ R^{n} be open. Let P be of the form (4.9) near \overline{W} and let ϕ be smooth and real near \overline{W}. Assume that \overline{W} is covered by finitely many open sets W_{1}, .., W_{N} such that (4.20) (or equivalently (4.21)) holds for u ∈ C_{0}^{∞}(Ω∩W_{j}) with u_{|}_{∂Ω} = 0. Then we shall see that it also holds for all u ∈ C_{0}^{∞}(Ω ∩ W) with u_{|}_{∂Ω} = 0.

Indeed, let 0 ≤ χ_{j} ∈ C_{0}^{∞}(W_{j}) with ∑_{1}^{N} χ_{j}^{2} = 1 near \overline{W} and apply (4.21) to χ_{j}u:

h ∫ e^{2ϕ/h}(|χ_{j}u|^{2} + |χ_{j}h∇_{x}u|^{2})dx ≤  (4.32)
O(1) ∫ e^{2ϕ/h}|χ_{j}P u|^{2}dx +O(1) ∫ e^{2ϕ/h}|[P, χ_{j}]u|^{2}dx
+O(1) ∫ e^{2ϕ/h}|[h∇_{x}, χ_{j}]u|^{2}dx.

Summing and using that [P, χ_{j}] = h ∑_{|α|≤1} b_{α,j}(x; h)hD_{x}, we get

h ∫ e^{2ϕ/h}(|u|^{2} + |h∇u|^{2})^{2}dx ≤ O(1) ∫ e^{2ϕ/h}|P u|^{2}dx +  (4.33)
O(1)h^{2} ∫ e^{2ϕ/h}(|u|^{2} + |h∇_{x}u|^{2})dx.

For h small enough, the last term can be absorbed by the first member of the inequality. #

4.4 Construction of ϕ satisfying the Poisson bracket condition.

Our function ϕ will satisfy

|dϕ| ≥ Const. > 0.  (4.34)

Write

p(x, ξ) = a(x, ξ) + p_{1}(x, ξ) + p_{0}(x), a(x, ξ) = ⟨A(x)ξ, ξ⟩.  (4.35)

We will replace ϕ by τϕ for τ > 0 large enough. Then

p_{τϕ}(x, ξ) = p(x, ξ + iτϕ′_{x}) =  (4.36)

a(x, ξ) − τ^{2}a(x, ϕ′) + 2iτ⟨A(x)ξ, ϕ′(x)⟩ + p_{1}(x, ξ) + iτp_{1}(x, ϕ′(x)) + p_{0}(x) = τ^{2}(q̃_{2}(x, ξ) + 2iq̃_{1}(x, ξ)),

with

q̃_{2}(x, ξ) = a(x, \frac{ξ}{τ}) − a(x, ϕ′_{x}) + \frac{1}{τ}p_{1}(x, \frac{ξ}{τ}) + \frac{1}{τ^{2}} p_{0}(x),

q̃_{1}(x, ξ) = ⟨A(x)\frac{ξ}{τ} , ϕ′(x)⟩ + \frac{1}{2τ} p_{1}(x, ϕ′(x)).

Let us look for the commons zeros of the real and imaginary parts: First notice that at such a common zero, we must have |ξ| ∼ τ|ϕ′(x)|, uniformly with respect to ϕ, for τ sufficiently large. Then we get

\{ \begin{matrix}a(x, \frac{ξ}{τ}) − a(x, ϕ′(x) = O_{ϕ}(1/τ) \\ ⟨A(x)\frac{ξ}{τ} , ϕ′(x)⟩ = O_{ϕ}(1/τ). \end{matrix}  (4.37)

At such common zeros, we compute

\frac{∂q̃_{2}}{∂ξ} = \frac{1}{τ}2A(x)\frac{ξ}{τ} + O_{ϕ}(\frac{1}{τ^{2}} ),
\frac{∂q̃_{2}}{∂x} = \frac{∂a}{∂x}(x, \frac{ξ}{τ}) − \frac{∂a}{∂x}(x, ϕ′(x)) − 2ϕ′′(x)A(x)ϕ′(x) + O_{ϕ}(\frac{1}{τ}),
\frac{∂q̃_{1}}{∂ξ} = \frac{1}{τ}A(x)ϕ′(x),
\frac{∂q̃_{1}}{∂x} = ϕ′′(x)A(x)\frac{ξ}{τ} + ⟨\frac{∂A}{∂x} \frac{ξ}{τ} , ϕ′(x)⟩ + O_{ϕ}(\frac{1}{τ}),

leading to

{q̃_{2}, q̃_{1}} =  (4.38)

= ⟨\frac{1}{τ}2A(x)\frac{ξ}{τ} , ϕ′′(x)A(x)\frac{ξ}{τ} ⟩ + ⟨\frac{1}{τ}2A(x)\frac{ξ}{τ} , ⟨\frac{∂A(x)}{∂x} \frac{ξ}{τ} , ϕ′(x)⟩⟩

−⟨\frac{∂a}{∂x}(x, \frac{ξ}{τ}) − \frac{∂a}{∂x}(x, ϕ′(x)) − 2ϕ′′(x)A(x)ϕ′(x), \frac{1}{τ}A(x)ϕ′(x)⟩ + O_{ϕ}(\frac{1}{τ^{2}} )

= \frac{2}{τ} [⟨A(x)\frac{ξ}{τ} , ϕ′′(x)A(x)\frac{ξ}{τ} ⟩ + ⟨A(x)ϕ′(x), ϕ′′(x)A(x)ϕ′(x)⟩ + O(|ϕ′(x)|^{3})]

+O_{ϕ}(\frac{1}{τ^{2}} ).

Now look for ϕ of the form

ϕ(x) = e^{βψ(x)},  (4.39)

where dψ(x) ≠ 0. Then

ϕ′(x) = e^{βψ(x)}βψ′(x),  (4.40)
ϕ′′(x) = e^{βψ(x)}β^{2}ψ′(x) ⊗^{t}ψ′(x) + e^{βψ(x)}βψ′′(x),  (4.41)

and (4.38) gives with ξ = τ|ϕ′(x)|η, |η| ∼ 1:

\{q̃_{2}, q̃_{1}\} = \frac{2}{τ}e^{3βψ}[β^{4}|ψ′(x)|^{2}⟨A(x)η, (ψ′(x) ⊗^{t}ψ′(x))A(x)η⟩  (4.42)

+β^{4}⟨A(x)ψ′(x), (ψ′(x) ⊗^{t}ψ′(x))A(x)ψ′(x)⟩ + O(β^{3})] + O_{β,ψ}(\frac{1}{τ^{2}} ).

Here ψ′(x) ⊗^{t}ψ′(x) ≥ 0 and

⟨A(x)ψ′(x), (ψ′(x) ⊗^{t}ψ′(x))A(x)ψ′(x)⟩ = ⟨A(x)ψ′(x), ψ′(x)⟩^{2} > 0,

so with β > 0 large enough, and for τ large enough depending on β, we get (4.15) with ϕ replaced by τϕ.

4.5 Estimates in a ball.

Recall the situation in Burq’s theorem: O ⊂⊂ R^{n} is open with smooth boundary, Ω = R^{n} \ O is connected, and P = ∑_{|α|≤2} a_{α}(x; h)D^{α} is a second order, elliptic, formally self-adjoint operator with smooth coefficients in C^{∞}(Ω). Moreover P is equal to −Δ outside some ball B(0, R_{0}), with R_{0} > 0. We may assume that \overline{O} ⊂ B(0, R_{0}). We also recall that we take the Dirichlet realization of P, i.e. we realize P as a self-adjoint operator L^{2} → L^{2} with domain H^{2} ∩ H_{0}^{1})(Ω). We are interested in estimates for P − λ^{2}, when λ → +∞. Put h = 1/λ, and write

(P − λ^{2}) = λ^{2}(h^{2}P − 1).  (4.43)

h^{2}P − 1 is then then of the type that we considered above.

Lemma 4.4 Let R_{2} > R_{0}. Then there exists ψ ∈ C^{∞}(\overline{B(0, R_{2})}\O) such that ψ = Const and \frac{∂ψ}{∂ν} > 0 on ∂O, and dψ(x) is ≠ 0 for all x ∈ \overline{B(0, R_{2})} \ O).

Proof. First, we can find a smooth real function ψ_{0} on \overline{B(0, R_{2})} \ O) such that ψ_{0} = 0 and \frac{∂ψ_{0}}{∂ν} > 0 on ∂O. Recall that if M is a smooth manifold and f : M → R is a smooth function, then x_{0} ∈ M is called a critical point, if df(x_{0}) vanishes. Such a critical point is non-degenerate if the Hessian f′′(x_{0}) which can be defined in local coordinates or more generally as a linear map from the tangent space T_{x_{0}}M into itself, is invertible. A Morse function is a smooth function f as above for which all critical points are non-degenerate. It is a well-known fact in differential topology that the Morse functions on M form a set which is dense in C^{∞}(M; R). Let ψ_{1} be a smooth real-valued extension of ψ_{0} to a neighborhood of \overline{B(0, R_{2})}\O, and let ψ_{2} be a Morse function on the same neighborhood which is very close to ψ_{0} in the norm of C^{1}(\overline{B(0, R_{2})}\O). Let ψ_{3} be the restriction of ψ_{2} to \overline{B(0, R_{2})} \ O. Then the critical points of ψ_{3} are of finite number and away from ∂O. We can modify ψ_{3} into a new smooth function ψ_{4} on \overline{B(0, R_{2})} \ O), which has the same critical points as ψ_{3} and such that ψ_{4}(x) = 0, \frac{∂ψ_{4}(x)}{∂ν} > 0 on ∂O. Let x_{1}, .., x_{N} ∈ \overline{B(0, R_{2})} \ O) be the critical points of ψ_{4}. Let [0, 1] ∋ t ↦ x_{j}(t) ∈ B(0, R_{2}+1)\\overline{O} be smooth curves with x_{j}(0) = x_{j}, x_{j}(1) ∈ ∂B(0, R_{2} + 1). Then we can construct a smooth family of diffeomorphisms κ_{t} : \overline{B(0, R_{2})} \ O → Ω_{t} \ O, where Ω_{t} ⊃ \overline{B(0, R_{2})}, such that κ_{t}(x) = x for x near ∂O and such that κ_{t}(x_{j}) = x_{j}(t). Then the lemma follows, if we let ψ be the restriction to \overline{B(0, R_{2})} \ O) of ψ_{4} ∘ κ_{1}^{−1}. #

From ψ, we form ϕ = τe^{βψ} as in section 4.4, and get the estimate:

h ∫ e^{2ϕ/h}(|u|^{2} + |h∇u|^{2})dx ≤ O(1) ∫ e^{2ϕ/h}|(h^{2}P − 1)u|^{2}dx,  (4.44)

for all u ∈ C_{0}^{∞}(B(0, R_{2}) \ O) with u_{|}_{∂O} = 0. Recalling that h = 1/λ, we get

λ^{3} ∫ e^{2λϕ}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx ≤ O(1) ∫ e^{2λϕ}|(P − λ^{2})u|^{2}dx.  (4.45)

Here we want to remove the assumption that u has compact support. Take R_{1} with R_{0} < R_{1} < R_{2}, and take χ with 1_{B(0,R_{1})} ≺ χ ∈ C_{0}^{∞}(B(0, R_{2})), and apply (4.45) to χu:

λ^{3} ∫_{B(0,R_{1})\O} e^{2λϕ}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx ≤  (4.46)

O(1) ∫_{B(0,R_{2})\O} e^{2λϕ}|(P − λ^{2})u|^{2}dx + O(1)λ^{2} ∫_{B(0,R_{1},R_{2})} e^{2λϕ}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx,

where B(0, R_{1}, R_{2}) = {x ∈ R^{n}; R_{1} < |x| < R_{2}} denotes the open shell of center 0 and radii R_{1}, R_{2}. To the left we estimate the exponential factor from below by e^{−λC} and to the left we estime the exponenential factors from above by e^{λC}, where C > 0 is a sufficiently large constant. It follows that there is a constant C > 0 such that

∫_{B(0,R_{1})\O}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx ≤  (4.47)

O(1)e^{2λC} ∫_{B(0,R_{2})\O} |(P − λ^{2})u|^{2}dx + O(1)e^{2λC} ∫_{B(0,R_{1},R_{2})}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx.

4.6 Estimate in a shell.

We work in the region |x| > R_{0}, where h^{2}P − 1 = −h^{2}Δ − 1 has the semiclassical symbol p = ξ^{2} − 1. In polar coordinates, x = rω, ω ∈ S^{n−1}, r > R_{0}, we get

p = ρ^{2} + \frac{(ω^{∗})^{2}}{r^{2}} − 1,

where (ω^{∗})^{2} ≥ 0 denotes the principal symbol of −Δ_{S^{n−1}}. We want to make Carleman estimates with radial weight ϕ = ϕ(r). We get

p_{ϕ} = (ρ + iϕ′)^{2} + \frac{1}{r^{2}} (ω^{∗})^{2} − 1,

so

\{ \begin{matrix}Re p_{ϕ} = ρ^{2} − (ϕ′)^{2} + \frac{(ω^{∗})^{2}}{r^{2}} − 1 \\ Im p_{ϕ} = 2ρϕ′, \end{matrix}  (4.48)

\frac{1}{4}\{Re p_{ϕ}, Im p_{ϕ}\} = ρ^{2}ϕ′′ + (ϕ′′ϕ′ + \frac{(ω^{∗})^{2}}{r^{3}} )ϕ′.

We need the last Poisson bracket to be > 0 on the joint characteristics where both expressions in (4.48) are equal to 0. We shall have ϕ′ > 0, so p_{ϕ} = 0 is equivalent to

ρ = 0, \frac{(ω^{∗})^{2}}{r^{2}} = 1 + (ϕ′)^{2}.  (4.49)

On this set, we need

ϕ′ > 0, ϕ′′ϕ′ + \frac{1 + (ϕ′)^{2}}{r} > 0.  (4.50)

With ψ = (ϕ′)^{2} we get the linear system of inequalities,

\frac{1}{2}ψ′ + \frac{1 + ψ}{r} > 0, ψ > 0.

This will follow from

ψ′ ≥ −\frac{2}{r}, ψ > 0.

Start by taking ψ = 2(A − log r), A ≫ 0, for 1 ≤ r ≤ B < e^{A}. Choosing B very close to e^{A}, we may have ψ(B) > 0 as small as we like. We can extend the definition of ψ to all r in such a way that ψ > 0 and

−\frac{2}{r} ≤ ψ′ ≤ 0,

with equality to the right for r ≥ e^{A}. Then ψ = ϵ^{2} for r ≥ e^{A} with ϵ > 0 as small as we like.

In conclusion, we can find a smooth real function ϕ on [1, +∞[ satisfying (4.50), and such that

ϕ′(r) = \sqrt{2(A − log r)}, 1 ≤ r ≤ e^{A} − 1,  (4.51)
ϕ′ is decreasing on [e^{A} − 1, +∞[ and equal to ϵ for r ≥ e^{A}.  (4.52)

Let R_{3} > R_{2}. Then for u ∈ C_{0}^{∞}(B(0, R_{0}, R_{3})) (the open shell), we have the Carleman estimate,

λ^{3} ∫ e^{2λϕ}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx ≤ O_{A,ϵ,R_{3},R_{0}}(1) ∫ e^{2λϕ}|(P − λ^{2})u|^{2}dx. (4.53)

4.7 Estimate in a ball ∪ shell.

We shall combine (4.47), valid for functions on B(0, R_{2}) \ O, with (4.53). We choose R_{1}, R_{2} so that R_{0} < R_{1} − 2 and R_{2} = R_{1} + 1. Choose ϕ as in section 4.6 and such that ϕ(R_{1}) − ϕ(R_{1} − 1) is larger than the constant C in the exponents in (4.47). Modifying ϕ by a constant, we get ϕ ≤ −ϵ_{0}, |x| ≤ R_{1} − 1, ϕ ≥ C + ϵ_{0}, |x| ≥ R_{1}, for some ϵ_{0} > 0.

Let u ∈ C^{∞}(B(0, R_{3}) \ O), u_{|}_{∂O} = 0, with

(P − λ^{2})u = v ∈ C_{0}^{∞}(B(0, R_{0}) \ O).  (4.54)

Then from (4.47), (4.53), we get:

∫_{B(0,R_{1})\O}(|u|^{2} + \frac{1}{λ}∇u|^{2})dx ≤  (4.55)

O(1)e^{2λC} ∫ |v|^{2}dx + O(1)e^{2λC} ∫_{B(0,R_{1},R_{2})}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx,

∫_{B(0,R_{1},R_{3}−1)} e^{2λϕ}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx ≤ O(1) ×  (4.56)

( ∫_{B(0,R_{1}−2,R_{1}−1)} e^{2λϕ}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx + ∫_{B(0,R_{3}−1,R_{3})} e^{2λϕ}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx).

There is a δ > 0 such that the last term of the right hand side of (4.55) can be estimated by e^{−λδ} times the left hand side of (4.56) and the first term of the right hand side of (4.56) can be estimated by e^{−λδ} times the left hand side of (4.55), when λ > 0 is large enough. So, if we sum (4.55), (4.56), we get rid of these two contributions to the the right hand sides and with

ϕ_{+}(x) = \{ \begin{matrix}0, |x| < R_{1}, \\ ϕ(x), |x| ≥ R_{1}, \end{matrix}

we get

∫_{B(0,R_{3}−1)\O} e^{2λϕ_{+}}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx ≤  (4.57)

O(1)e^{2λC} ∫ |v|^{2}dx + O(1) ∫_{B(0,R_{3}−1,R_{3})} e^{2λϕ_{+}}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx,

for u, v as in (4.54).

4.8 Estimates for outgoing solutions.

In the previous estimate, we may assume that R_{3} is as large as we like, and that ϕ(x) = Const. + ϵ|x|, in B(0, \frac{R_{3}}{4} , ∞) with ϵ as small as we like. (The choice of ϵ, R_{3}, will only affect the ”O(1)” factors and the required largeness of λ, but not the constant C in the exponent.) Let u be as before, but assume in addition that u is outgoing: u = R(λ)v, where v ∈ L_{comp}^{2}(B(0, R_{0}) \ O). For 1_{supp v∪\overline{O}} ≺ χ ∈ C_{0}^{∞}(R^{n}; [0, 1]), we have

(\frac{1}{i}[P, χ]u|u) = \frac{1}{i}((χu|P u)) − (P u|χu))  (4.58)
= \frac{λ^{2}}{i} ((χu|u) − (u|χu)) + \frac{1}{i}((χu|v) − (v|χu))
= \frac{1}{i}((u|v) − \overline{(u|v)}) = 2Im (u|v).

By the Cauchy–Schwartz inequality, we have

(\frac{1}{i}[P, χ]u|u) ≤ 2∥u∥_{B(0,R_{0})}∥v∥_{B(0,R_{0})}.  (4.59)

(Recall that u is outgoing, so that u = R_{0}(λ)w for |x| ≥ R_{0}, for some

w ∈ C_{0}^{∞}(B(0, R_{0}) \ \overline{O}).

Then by Proposition 2.5

(\frac{1}{i}[P, χ]u|u) = C(λ) ∫_{∂B(0,λ)} |ŵ(ξ)|^{2}L_{∂B(0,λ)}(dξ),

where C(λ) > 0 and we assume that λ > 0 in order to fix the ideas. So the scalar product in (4.59) is ≥ 0.)

The semi-classical principal symbol of \frac{1}{i}[hP, χ] is −H_{p}(χ) = −2ξ ·∂_{x}χ(x), which is ≥ 0 on the outgoing trajectories (or ”rays”). Here by an outgoing ray, we mean a maximal(ly extended) integral curve (necessarily a line segment) of H_{p} = 2ξ · ∂_{x} in

\{(x, ξ) ∈ R^{2n}; ξ^{2} = 1, \frac{R_{3}}{2} < |x| < 2R_{3}\},  (4.60)

such that the backward extension (for negative times, as an integral curve of H_{ξ^{2}}), passes over B(0, \frac{R_{3}}{4} ). By an incoming ray, we mean a maximal ray in the set (4.60) which is not an outgoing one.

If we work in the shell \frac{R_{3}}{2} ≤ |x| ≤ 2R_{3}, and consider relative sizes compared to ∥u∥_{H^{1}(B(0, \frac{R_{3}}{4} , \frac{R_{3}}{3} ))}, then by using phase space analysis, one can show

that ”u is O(e^{−λR_{3}/C})” near the incoming trajectories (this is a consequence of the fact that u is outgoing, see [38]) and ”O_{δ}(e^{δλR_{3}})” for every δ > 0, near the outgoing trajectories. Also u is exponentially localized to the energy surface ξ^{2} = 1, when we restrict x to a shell as above. In view of these facts, one expects (and can indeed show) that (\frac{1}{i}[P, χ]u|u) behaves like the square of a norm of u. More precisely we have ([17]) for every δ > 0 and for λ ≥ λ(δ):

∫_{B(0,R_{3}−1,R_{3})}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx ≤  (4.61)

e^{−2λR_{3}/C_{0}} ∫_{B(0, \frac{R_{3}}{4} , \frac{R_{3}}{3} )}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx + e^{2R_{3}δλ}(\frac{1}{i}[P, χ]u|u),

where C_{0} > 0. Combining this with (4.57), (4.59) leads to

∫_{B(0,R_{3}−1)\O} e^{2λϕ_{+}}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx ≤  (4.62)

O(1)e^{2λC} ∫ |v|^{2}dx + O(1)e^{2λ(ϕ_{+}(R_{3})− \frac{R_{3}}{C_{0}} )} ∫_{B(0, \frac{R_{4}}{4} , \frac{R_{3}}{3} )}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx

+O(1)e^{2λ(ϕ_{+}(R_{3})+δ)}∥u∥_{B(0,R_{0})}∥v∥_{B(0,R_{0})}.

Since |∇ϕ| ≤ ϵ for \frac{R_{3}}{4} ≤ |x|, the second term to the right can be absorbed by the left hand side. So can the u-contribution to the last term, if we use the standard inequality ”ab ≤ \frac{ϵ}{2}a^{2} + \frac{1}{2ϵ}b^{2}”:

O(1)e^{2λ(ϕ_{+}(R_{3})+δ)}∥u∥ ≤ \frac{1}{2}∥u∥^{2} + O(1)e^{4λ(ϕ_{+}(R_{3})+δ)}∥v∥^{2}.

With a new constant C depending on R_{3}, we obtain:

∫_{B(0,R_{3}−1)\O}(|u|^{2} + |\frac{1}{λ}∇u|^{2})dx ≤ e^{2λC} ∫ |v|^{2}dx.  (4.63)

This implies (4.4) and concludes our outline of the proof of Burq’s theorem.

5 Review of non-selfadjoint spectral theory

[Missing piece: Fredholm theory via Grushin problems as in an appendix of [38].]

In this chapter we review some of the standard theory for non-selfadjoint operators, and we will essentially follow the book [31]. Let H be a separable complex Hilbert space. If A ∈ L(H, H) is compact, we let s_{1}(A) ≥ s_{2}(A) ≥ ... ↘ 0 be the eigenvalues of (A^{∗}A)^{1/2}. They are called the singular values of A. We notice that s_{j}(A^{∗}) = s_{j}(A). In fact, this follows from the intertwining relations:

A(A^{∗}A) = (AA^{∗})A, (A^{∗}A)A^{∗} = A^{∗}(AA^{∗}).

The singular values appear naturally in the polar decomposition: If A ∈ L(H, H), then

∥Au∥^{2} = (Au|Au) = (A^{∗}Au|u) = ((A^{∗}A)^{1/2}u|(A^{∗}A)^{1/2}u) = ∥(A^{∗}A)^{1/2}u∥^{2}.

The operator

U : R((A^{∗}A)^{1/2}) ∋ (A^{∗}A)^{1/2}u ↦ Au ∈ R(A)

is isometric and bijective. It extends to a unitary operator that we also denote by U from \overline{R((A^{∗}A)^{1/2}} to \overline{R(A)} and to a partial isometry if we put U = 0 on the orthogonal space (R(A^{∗}A)^{1/2})^{⊥} = N ((A^{∗}A)^{1/2}) = N (A), we get the polar decomposition :

A = U(A^{∗}A)^{1/2}.  (5.1)

This leads to the Schmidt decomposition of A: Let e_{1}, e_{2}, .. be an orthonomral family of eigenvectors of (A^{∗}A)^{1/2} associated to the eigenvalues s_{1}(A), s_{2}(A), .. that are > 0. Then

Au = ∑ s_{j}(A)(u|e_{j})f_{j},  (5.2)

where f_{j} = Ue_{j} is also an orthonormal family.

Recall the mini-max characterization of the the s_{j}:

s_{j}(A) = inf_{E⊂H; E is a closed subspace of H of codimension ≤j−1} sup_{u∈E\0} \frac{((A^{∗}A)^{1/2}u|u)}{∥u∥^{2}}.  (5.3)

From that we get the following characterization of the singular values which is due to Allaverdiev:

Theorem 5.1 Let A ∈ L(H, H) be compact. Then

s_{n+1}(A) = min_{{K∈L(H,H) \atop K of rank ≤n}} ∥A − K∥, n = 0, 1, .. .

The minimum is realized by an operator K for which s_{1}(K) = s_{1}(A),..,s_{n}(K) = s_{n}(A), s_{n+1}(K) = 0, s_{1}(A − K) = s_{n+1}(A), s_{2}(A − K) = s_{n+2}(A),.. .

Proof. If K is of rank ≤ n, then N (K) is of codimension ≤ n and

s_{n+1}(A) ≤ sup_{0≠u∈N (K)} \frac{∥Au∥}{∥u∥} = sup_{0≠u∈K} \frac{∥(A − K)u∥}{∥u∥} ≤ ∥A − K∥.

To get the minimizing operator write the polar decomposition A = U(A^{∗}A)^{1/2} and take K = U(A^{∗}A)^{1/2}P_{n}, where P_{n} is the orthogonal projection onto the space spanned by e_{1}, .., e_{n}. Then

A − K = U(A^{∗}A)^{1/2}(1 − P_{n}),

(A − K)^{∗}(A − K) = (1 − P_{n})(A^{∗}A)^{1/2}U^{∗}U(A^{∗}A)^{1/2}(1 − P_{n}) = (A^{∗}A)(1 − P_{n}),

and we get the statement about the singular values of A − K. Especially s_{n+1}(A) = ∥A − K∥. The statement about the singular values of K can be obtained similarly. #

The following corollary is due to Ky Fan:

Corollary 5.2 Let A, B ∈ L(H) be compact. Then for n, m ≥ 0:

s_{m+n−1}(A + B) ≤ s_{m}(A) + s_{n}(B)  (5.4)
s_{m+n−1}(AB) ≤ s_{m}(A)s_{n}(B).  (5.5)

Proof. Let K_{A}, K_{B} be operators of rank ≤ m − 1 and ≤ n − 1 respectively, such that

s_{m}(A) = ∥A − K_{A}∥, s_{n}(B) = ∥B − K_{B}∥.

Then

s_{m+n−1}(A+B) ≤ ∥A+B−(K_{A}+K_{B})∥ ≤ ∥A−K_{A}∥+∥B−K_{B}∥ = s_{m}(A)+s_{n}(B).

The proof for AB is essentially the same. #

Corollary 5.3 We have |s_{n}(A) − s_{n}(B)| ≤ ∥A − B∥.

Proof. Let K be an operator of rank n − 1. Then

s_{n}(A) ≤ ∥A − K∥ = ∥B − K + A − B∥ ≤ ∥B − K∥ + ∥A − B∥.

Varying K, we get s_{n}(A) ≤ s_{n}(B)+∥A−B∥, and we have the same inequality with A and B exchanged. #

We now discuss Weyl inequalities, and start with the following result to H. Weyl:

Theorem 5.4 Let A ∈ L(H, H) be compact and let λ_{1}(A), λ_{2}(A), .. be the non-vanishing eigenvalues of A arranged in such a way that |λ_{1}| ≥ |λ_{2}| ≥ ... and repeated according to their multiplicity (which by definition is the rank of the spectral projection). The for every n ≥ 1 for which λ_{n}(A) is defined, we have

|λ_{1}(A) · .. · λ_{n}(A)| ≤ s_{1}(A) · .. · s_{n}(A).  (5.6)

(For notational reasons, we assume that there are infinitely many non-vanishing eigenvalues of A.)

Proof. For n = 1, (5.6) just says that |λ_{1}(A)| ≤ ∥A∥. Approaching A by a sequence of finite rank operators, we can assume that A is of finite rank and replace H by the finite dimensional space R(A) + (N (A))^{⊥}, that we denote by H from now on. [[[Haer maaste vi saega naagot om kontinueten av egenvaerden som funktioner av A.]]] Introduce the space

∧^{n}H = H ∧ ... ∧ H  (5.7)

generated by n-fold exterior products of vectors in H. ∧^{n}H is a Hilbert space with a scalar product that satisfies

(u_{1} ∧ .. ∧ u_{n}|v_{1} ∧ .. ∧ v_{n}) = det((u_{j}|v_{k})), u_{j}, v_{j} ∈ H.  (5.8)

Further, there is a linear operator ∧^{n}A : ∧^{n}H → ∧^{n}H which is uniquely determined by the condition,

(∧^{n}A)(u_{1} ∧ .. ∧ u_{n}) = Au_{1} ∧ .. ∧ Au_{n}, u_{j} ∈ H.  (5.9)

Using a basis of generalized eigenvectors, we see that the eigenvalues of ∧^{n}A are the values λ_{j_{1}} · .. · λ_{j_{n}}, with j_{ν} ≠ j_{μ}, for ν ≠ μ. The eigenvalue of greatest modulus is then λ_{1} · .. · λ_{n}. On the other hand the adjoint of ∧^{n}A is ∧^{n}A^{∗}. We also have (∧^{n}A)(∧^{n}B) = ∧^{n}(AB). Then (∧^{n}A)^{∗}(∧^{n}A) = ∧^{n}(A^{∗}A) and this operator has the eigenvalues s_{j_{1}}(A)·..·s_{j_{n}}(A)^{2} out of which the largest one is

(s_{1}(A) · .. · s_{n}(A))^{2} = ∥ ∧^{n} A∥^{2} ≥ |λ_{1}| · .. · |λ_{n}|^{2}.

The proof is complete. #

In the same spirit we have the inequality of A. Horn:

∏_{1}^{n} s_{j}(AB) ≤ (∏_{1}^{n}(s_{j}(A)s_{j}(B))  (5.10)

Proof. As before it suffices to treat the case when H is of finite dimension. The largest eigenvalue of

(∧^{n}AB)^{∗}(∧^{n}AB) = ∧^{n}((AB)^{∗}AB)

is equal to (s_{1}(AB) · .. · s_{n}(AB))^{2}. On the other hand,

((∧^{n}AB)^{∗}(∧^{n}AB)u|u) = ∥(∧^{n}AB)u∥^{2} = ∥(∧^{n}A) ∘ (∧^{n}B)u∥^{2}

≤ ∥ ∧^{n} A∥^{2}∥ ∧^{n} B∥^{2}∥u∥^{2} ≤ (s_{1}(A) · .. · s_{n}(A))^{n}(s_{1}(B) · .. · s_{n}(B))^{2}∥u∥^{2},

and taking the supremum over all normalized u, we obtain the required inequality. #

We next need a convexity inequality, due to Weyl and Littlewood–Polya.

Lemma 5.5 Let Φ(x) be a convex function on R, which tends to 0, when x → −∞, with Φ(−∞) := lim_{x→−∞} Φ(x) = 0. Let a_{1} ≥ .. ≥ a_{N}, b_{1} ≥ .. ≥ b_{N} be real numbers with

∑_{1}^{k} a_{j} ≤ ∑_{1}^{k} b_{j}, 1 ≤ k ≤ N.

Then,

∑_{1}^{k} Φ(a_{j}) ≤ ∑_{1}^{k} Φ(b_{j}), 1 ≤ k ≤ N.

Proof. Approaching Φ by a sequence of smooth functions, we can reduce the proof to the case when Φ ∈ C^{∞}. Then Φ′ ≥ 0,

Φ′(x) → 0, x → −∞.  (5.11)

Letting y → −∞ in the identity

Φ(x) = Φ(y) + ∫_{y}^{x} Φ′(t)dt,  (5.12)

we get

Φ(x) = ∫_{−∞}^{x} Φ′(t)dt.

From the convergence of the last integral, we conclude that ∫_{y}^{0} Φ′(t)dt ≤ C, implying that |y|Φ′(y), is a bounded function for y ≤ 0, which tends to 0 when y → −∞.

Integration by parts in (5.12) gives

Φ(x) = Φ(y) + [(t − x)Φ′(t)]_{t=y}^{x} − ∫_{y}^{x}(t − x)Φ′′(t)dt
= Φ(y) + (x − y)Φ′(y) + ∫_{y}^{x}(x − t)Φ′′(t)dt.

Letting y tend to −∞, we get

Φ(x) = ∫_{−∞}^{x}(x − t)Φ′′(t)dt = ∫ (x − t)_{+}Φ′′(t)dt.

Hence

∑_{1}^{k} Φ(a_{j}) = ∫ ( ∑_{j =1}^{k}(a_{j} − t)_{+})Φ′′(t)dt,

for every t. Let k(t) ≤ k be the largest k̃ ≤ k with a_{k} ≥ t. Then

∑_{j=1}^{k}(a_{j} − t)_{+} = ∑_{j =1}^{k(t)}(a_{j} − t) ≤ ∑_{j=1}^{k(t)}(b_{j} − t) ≤ ∑_{j=1}^{k}(b_{j} − t)_{+}.

Hence ∑_{1}^{k} Φ(a_{j}) ≤ ∑_{1}^{k} Φ(b_{k}). #

As a consequence we get the following result of Weyl :

Theorem 5.6 Let A : H → H be a compact operator, and f(x) ≥ 0 a function on [0, ∞[ with f(0) = 0 such that f(e^{t}) is convex. Let λ_{j} and s_{j} be the eigenvalues and singular values of A, arranged with |λ_{1}| ≥ |λ_{2}| ≥ ..., s_{1} ≥ s_{2} ≥ .... Then for every k ≥ 1 :

∑_{1}^{k} f(|λ_{j}|) ≤ ∑_{1}^{k} f(s_{j}).  (5.13)

Proof. We know that

∑_{1}^{k} log |λ_{j}| ≤ ∑_{1}^{k} log s_{j},

and it suffices to apply the preceding convexity lemma. #

Corollary 5.7 For every p > 0, we have

∑_{1}^{n} |λ_{j}(A)|^{p} ≤ ∑_{1}^{n} s_{j}(A)^{p}.

For every r > 0, we have

∏_{1}^{n}(1 + r|λ_{j}(A)|) ≤ ∏_{1}^{n}(1 + rs_{j}(A)).

Let A, B be compact operators. With Φ(t) = e^{t}, a_{j} = log s_{j}(AB), b_{j} = log(s_{j}(A)s_{j}(B)), we get from Horn’s inequality (5.10) and Lemma 5.5 :

Corollary 5.8 ∑_{1}^{n} s_{j}(AB) ≤ ∑_{1}^{n} s_{j}(A)s_{j}(B).

Let C_{∞} ⊂ L(H) be the subspace of compact operators. The following Lemma is due to Ky Fan.

Lemma 5.9 Let A ∈ C_{∞}. Then for every 1 ≤ n ∈ N, we have

∑_{1}^{n} s_{j}(A) = max ∑_{j=1}^{n}(UAϕ_{j}|ϕ_{j}),

where the maximum is taken over the set of all unitary operators U and all orthonormal systems ϕ_{1}, .., ϕ_{n}.

We leave the proof as an exercise, or else see [31].

Corollary 5.10 If A, B ∈ S_{∞}, then

∑_{1}^{n} s_{j}(A + B) ≤ ∑_{1}^{n} s_{j}(A) + ∑_{1}^{n} s_{j}(B).

We are no ready to discuss the Schatten–von Neumann classes.

Definition. For 1 ≤ p ≤ ∞, we put

C_{p} = {A ∈ C_{∞}; ∑_{1}^{∞} s_{j}(A)^{p} < ∞}.

Theorem 5.11 C_{p} is a closed two-sided ideal in L(H, H) equipped with the norm

∥A∥_{C_{p}} = ∥(s_{j}(A))_{1}^{∞}∥_{ℓ^{p}}.

If p_{1} ≤ p_{2}, then C_{p_{1}} ⊂ C_{p_{2}}. The space of finite rank operators is dense in C_{p} for every p.

We will only recall the proof of the fact that ∥ · ∥_{C_{p}} satisfies the triangle inequality. Let A, B ∈ C_{p}, and put ξ_{j} = s_{j}(A + B), η_{j} = s_{j}(A) + s_{j}(B). According to Corollary 5.10, we have ∑_{1}^{n} ξ_{j} ≤ ∑_{1}^{n} η_{j}, ∀n and hence,

∥A + B∥_{C_{1}} ≤ ∥A∥_{C_{1}} + ∥B∥_{C_{1}},

by letting n tend to ∞.

It remains to treat the case p > 1. ξ_{j} and η_{j} are both decreasing sequencies. It suffices to show that

∥ξ∥_{ℓ^{p}} ≤ ∥η∥_{ℓ^{p}}.  (5.14)

We have

∥ξ∥_{ℓ^{p}} = sup_{{ζ∈ℓ^{q}, \atop ∥ζ∥_{ℓ^{q}} =1}}⟨ξ, ζ⟩,

by Hölder, where q ∈ [1, +∞[ is the conjugate index, given by p^{−1} + q^{−1} = 1 and ⟨·, ·⟩ denotes the real scalar product on ℓ^{2}({1, 2, ..}). We also know that the supremum is attained by a ζ = ζ^{0} of the form ζ_{j}^{0} = (Const. > 0)ξ_{j}^{p/q} and in particular ζ_{j}^{0} is a decreasing sequence. We use partial summation with Ξ_{j} = ∑_{1}^{j} ξ_{k}, Ξ_{0} = 0:

⟨ξ, ζ^{0}⟩^{(n)} : = ∑_{1}^{n} ξ_{j}ζ_{j}^{0} = ∑_{j=1}^{n}(Ξ_{j} − Ξ_{j−1})ζ_{j}^{0}

= ∑_{j=1}^{n} Ξ_{j}ζ_{j}^{0} − ∑_{j=1}^{n−1} Ξζ_{j+1}^{0} = Ξ_{n}ζ_{n}^{0} + ∑_{j=1}^{n−1} Ξ_{j}(ζ_{j}^{0} − ζ_{j+1}^{0})

= (∑_{1}^{n} ξ_{k})ζ_{n}^{0} + ∑_{j=1}^{n−1} ∑_{k=1}^{j} ξ_{k}) \underbrace{(ζ_{j}^{0} − ζ_{j+1}^{0})}_{≥0}

The last expression is ≤ the same expression with ξ replaced by η and running the same calculation backwards the latter expression is equal to ⟨η, ζ^{0}⟩^{(n)} Hence ⟨ξ, ζ^{0}⟩^{(n)} ≤ ⟨η, ζ^{0}⟩ ≤ ∥η∥_{ℓ^{p}}. Letting n tend to infinity, we get (5.14), and this completes the proof of the triangle-inequality for the C_{p}-norms. #

We notice that ∥A∥ = s_{1}(A) ≤ ∥A∥_{C_{p}}. The space C_{1} is the space of nuclear or trace-class operators, and C_{2} is the space of Hilbert-Schmidt operators. We have the following Hölder type result :

Theorem 5.12 Let p, q ∈ [1, ∞] be conjugate indices; p^{−1} + q^{−1} = 1. If A ∈ C_{p}, B ∈ C_{q}, then AB ∈ C_{1} and ∥AB∥_{C_{1}} ≤ ∥A∥_{C_{p}}∥B∥_{C_{q}}.

Proof. We know that

∑_{1}^{n} s_{j}(AB) ≤ ∑_{1}^{n} s_{j}(A)s_{j}(B)

and letting n tend to ∞, we get from the usual Hölder inequuality:

∑_{1}^{∞} s_{j}(AB) ≤ ∑_{1}^{∞} s_{j}(A)s_{j}(B) ≤ ∥s_{·}(A)∥_{ℓ^{p}}∥s_{·}(B)∥_{ℓ^{q}} = ∥A∥_{C_{q}}∥B∥_{C_{q}}.

#

We next discuss the trace of a nuclear operator. If A ∈ L(H, H) is of finite rank, we choose a finite dimensional subspace H′ ⊂ H such that R(A) ⊂ H′, (H′)^{⊥} ⊂ N (A) We then define the trace of A, tr A as the trace tr A_{|}_{H′} of the restriction of A to H′. We check that this does not depend on the choice of H′ and that tr A is the sum of the finitely many non-vanishing eigenvalues of A (each counted with its algebraic multiplicity). We see that

A ↦ tr A  (5.15)

is a linear functional on the space of finite rank operators. Moreover,

|tr A| ≤ ∑ |λ_{j}(A)| ≤ ∥A∥_{C_{1}}.  (5.16)

We can then extend (5.15) to a continuous linear functional on C_{1} and we still have

|tr A| ≤ ∥A∥_{C_{1}}.  (5.17)

In the case of finite rank operators, we also have

tr AB = tr BA.  (5.18)

Let now A ∈ C_{p}, B ∈ C_{q}, where p, q ∈ [1, ∞] are conjugate indices and choose A_{ν}, B_{ν}, ν = 1, 2, .. of finite rank, so that ∥A − A_{ν}∥_{C_{p}} → 0, ∥B − B_{ν}∥_{C_{q}} → 0. Then

∥AB − A_{ν}B_{ν}∥_{C_{1}} = ∥(A − A_{ν})B + A_{ν}(B − B_{ν})∥_{C_{1}}
≤ ∥A − A_{ν}∥_{C_{p}}∥B∥_{C_{q}} + ∥A_{ν}∥_{C_{p}}∥B − B_{ν}∥_{C_{q}} → 0, ν → ∞.

Using this also for BA and the cyclictiy of the trace (5.18) for finite rank operators, we obtain it also in the case A ∈ C_{p}, B ∈ C_{q}, where p, q ∈ [1, ∞] are conjugate indices.

Remark. There is a simple way of extending most of the theory to the case of operators A : H_{1} → H_{2}, where H_{1}, H_{2} are two different Hilbert spaces. Consider namely the corresponding operator

( \begin{matrix}0 & 0 \\ A & 0 \end{matrix} ) : H_{1} ⊕ H_{2} → H_{1} ⊕ H_{2},  (5.19)

and say that A belongs to C_{p} if the operator in (5.19) does. We leave these extensions as an exercise for the reader, and notice simply that the cyclicity of the trace still holds in this setting, namely if A : H_{1} → H_{2}, and B : H_{2} → H_{1} belong to C_{p} and C_{q} respectively, where p and q are conjugate indices.

We next discuss determinants of trace-class perturbations of the identity operator. Let first A ∈ L(H, H) be of finite rank and chose a finite dimensional Hilbert space as above. Then we define

det(1 − A) = det((1 − A)_{|}_{H′}) = ∏_{j}(1 − λ_{j}(A)),  (5.20)

where λ_{j}(A) denote the non-vanishing eigenvalues, repeated according to their multiplicity. We remark that

| det(1 − A)| ≤ ∏_{j}(1 + |λ_{j}(A)|) ≤ ∏_{j}(1 + s_{j}(A)) ≤ e^{∑_{j} s_{j}(A)},

where the the second inequlity follows from Corollary 5.7 . We want to extend the definition to the case when A ∈ C_{1}. Let first I be a compact interval and let I ∋ t ↦ A_{t} be a C^{1}family of finite rank rank operators, with R(A_{t}) ⊂ H′, N (A_{t}) ⊂ (H′)^{⊥} for some finite dimensional subspace H′ which is independent of t. We first assume that 1 − A_{t} is invertible for all t ∈ I, or in other words that 1 − λ_{j}(A_{t}) ≠ 0 for all t and j. Then det(1 − A_{t}) ≠ 0 and by a classical formula,

\frac{∂}{∂t} log det(1 − A_{t}) = −tr ((1 − A_{t})^{−1} \frac{∂}{∂t}(A_{t})) = −tr ((\frac{∂}{∂t}A_{t})(1 − A_{t})^{−1}).

Hence,

| \frac{\frac{∂}{∂t} det(1 − A_{t})}{det(1 − A_{t})}| = |\frac{∂}{∂t} log det(1 − A_{t})| ≤ ∥(1 − A_{t})^{−1}∥∥\frac{∂}{∂t}A_{t}∥_{C_{1}}.

In particular, if I = [0, 1], A_{t} = tA_{1} + (1 − t)A_{0}, we get

| log det(1−A_{1})−log det(1−A_{0})| ≤ sup_{0≤t≤1} ∥(1−(tA_{1}+(1−t)A_{0}))^{−1}∥∥A_{1}−A_{0}∥_{C_{1}}.

(5.21)

Now let A ∈ C_{1}. If 1−A_{1} is not invertible, we put det(1−A) = 0. Assume then that 1 − A is invertible. Let A_{ν} be a sequence of finite rank operators which converges to A in C_{1}. For ν large enough, we have ∥(1 − A_{ν})^{−1}∥ ≤ C_{0} for some fixed constant C_{0} and more generally ∥(1 − (tA_{ν} + (1 − t)A_{0}))^{−1}∥ ≤ C_{0}. Then

| log det(1 − A_{ν}) − log det(1 − A_{μ})| ≤ C_{0}∥A_{ν} − A_{μ}∥_{C_{1}}

and consequently lim_{ν→∞}(log det(1 − A_{ν}) − log det(1 − A_{μ})) exists. We then put

det(1 − A) = det(1 − A_{μ}) exp lim_{ν→∞}(log det(1 − A_{ν}) − log det(1 − A_{μ})). (5.22)

Notice that

| det(1 − A)| ≤ ∏(1 + s_{j}(A)) ≤ e^{∥A∥^{C_{1}}}.  (5.23)

Using approximation by finite rank operators, we also see that

det((1 − A)(1 − B)) = det(1 − A) det(1 − B).  (5.24)

By the same argument, we can extend (5.21) to general trace clacc operators for which 1 − (tA_{1} + (1 − t)A_{0}) is invertible.

We now add a complex variable z ∈ C and consider the function det(1 − zA). If A is of finite rank, then this is an entire function of z. If A ∈ C_{1}, let A_{ν} → A be a sequence of finite rank operators. Then | det(1 − zA_{ν})| ≤ e^{|z|∥A_{ν}∥_{C_{1}}}. If zλ_{j}(A) ≠ 1, ∀j, then det(1 − zA_{ν}) → det(1 − zA) with locally uniform convergence in C \ ∪_{j}{1/λ_{j}}, and it follows that det(1 − zA) is a holomorphic function on this set, which verifies

| det(1 − zA)| ≤ e^{∥A∥_{C_{1}}|z|}.  (5.25)

It follows that det(1 − zA_{ν}) converges to an entire function f(z) locally uniformly on C. If z = 1/λ_{j}(A) where λ_{j}(A) is of multiplicity m, then exacly m eigenvlues of A_{ν} will converge to λ_{j}(A) while the others will stay away from a neighborhood of this point (when ν is large enough). Considering the argument variation (Rouché), we conclude that f(z) vanishes to the order m at 1/λ_{j}(A) and in particular we have f(z) = det(1 − zA) also at that point. In conclusion, we have

Proposition 5.13 Let A ∈ C_{1}. Then D_{A}(z) := det(1 − zA) is an entire function whose zeros counted with multiplicity coincide with the values 1/λ_{1}(A), 1/λ_{2}(A), .. counted with the multiplicities of λ_{1}(1), λ_{2}(A), ...

Observe that

D_{A}(0) = 1  (5.26)

Also observe that D_{A}(z) is of subexponental growth in the sense that for every ϵ > 0 there exists a constant C_{ϵ} > 0 such that

|D_{A}(z)| ≤ C_{ϵ}e^{ϵ|z|}.  (5.27)

In fact, by a limiting argument, we have

|D_{A}(z)| ≤ ∏_{1}^{∞}(1 + |z|s_{j}(A)) ≤ ∏_{1}^{N}(1 + |z|s_{j}(A))e^{∑_{N+1}^{∞} s_{j}(A))|z|}.

Here the prefactor is of polynomial growth for every fixed N and for a given ϵ > 0, we can always choose N > 0 so large that the exponent is ≤ ϵ|z|.

The next observation is that W (z) = ∏_{1}^{∞}(1 − zλ_{j}(A)) is also an entire function of subexponential growth. This follows by the same argument, if we recall that ∑_{1}^{∞} |λ_{j}(A)| is convergent. We now use a special case of a theorem of Hadamerd (see [2]): Since D_{A}(z) and W (z) have the same zeros (counted with multiplicity), we have

D_{A}(z) = W (z)e^{g(z)},

where g(z) is an entire function. It is also clear that we can choose g with g(0) = 0. The function

Re g(z) := log |D_{A}(z)| − ∑_{1}^{∞} log |1 − λz|  (5.28)

is harmonic. Let R ≥ 2. For |z| ≤ R/2, we have

Re g(z) = ∫_{|w|=R} P_{R}(z, w)Re g(w)ℓ(dw),  (5.29)

where P_{R} is the Poisson kernel for the disc of radius R and ℓ denotes the length element on the boundary of this disc. By a scaling argument, it is easy to see that

\frac{1}{CR} ≤ P_{R}(z, w) ≤ \frac{C}{R}, |z| ≤ \frac{R}{2}, |w| = R,  (5.30)

where C > 0 is independent of R. Using the subexponential growth of D_{A}(z), we get

∫_{|w|=R} P_{R}(z, w) log |D_{A}(w)|ℓ(dw) ≤ \frac{C}{R}ϵRR ≤ CϵR,  (5.31)

for R ≥ R_{ϵ} large enough. On the other hand,

| ∫_{|w|=R} P_{R}(z, w) ∑_{1}^{∞} log |1 − λ_{j}w|ℓ(dw)| ≤ ∑_{1}^{∞} \frac{1}{R} ∫_{|w|=R} | log |1 − λ_{j}w||ℓ(dw)

(5.32)

If |λ_{j}|R ≤ \frac{1}{2}, we write | log |1 − λ_{j}w|| ≤ C|λ_{j}|R and the sum over the corresponding js in (5.32) can be bounded by

∑_{|λ_{j}|≤ \frac{1}{2R}} \frac{C}{R}2π|λ_{j}|RR = 2πCR ∑_{|λ_{j}|≤ \frac{1}{2R}} |λ_{j}| = o(R), R → ∞.  (5.33)

If \frac{1}{2} ≤ |λ_{j}|R ≤ T, where T ≫ 1 is independent of R, then with a = |λ_{j}|R ∈ [\frac{1}{2}, T]:

\frac{1}{R} ∫_{|w|=R} | log |1 − λ_{j}w||ℓ(dw) = ∫_{|ζ|=1} | log |1 − aζ||ℓ(dζ).

With ζ = e^{iθ}, we get

|1 − aζ|^{2} = (1 − a cos θ)^{2} + a^{2}(sin θ)^{2}
= 1 + a^{2} − 2a cos θ = a(\frac{1}{a} + a − 2 cos θ) ≥ 2a(1 − cos θ).

Consequently,

\frac{1}{R} ∫_{|w|=R} | log |1 − λ_{j}w||ℓ(dw) ≤ C_{T} .

Let us estimate the number of λ_{j} in this case:

∑_{\frac{1}{2} ≤|λ_{j}|R≤T} 1 ≤ 2R ∑_{\frac{1}{2R} ≤|λ_{j}| \frac{T}{R}} |λ_{j}| = o_{T} (R).

Hence

∑_{\frac{1}{2} ≤|λ_{j}|R≤T} \frac{1}{R} ∫_{|w|=R} | log |1 − λ_{j}w||ℓ(dw) = o_{T} (R), R → ∞.  (5.34)

It remains to consider the case |λ_{j}|R ≥ T. Here log |1−λ_{j}R| ∼ log(|λ_{j}|R). Hence, with constants C and C_{δ} that are independent of T:

∑_{|λ_{j}|R≥T} \frac{1}{R} ∫_{|w|=R} | log |1 − λ_{j}w||ℓ(dw) ≤ C ∑_{|λ_{j}|R≥T} log(|λ_{j}|R)

≤ C_{δ} ∑_{|λ_{j}|≥ \frac{T}{R}} |λ_{j}|^{δ}R^{δ} = C_{δ}R^{δ} ∑_{|λ_{j}|≥ \frac{T}{R}} |λ_{j}|^{δ}.

Put δ = 1/p, 1 < p < ∞, and let q be the conjugate index. Then, if N denotes the number of λ_{j} with |λ_{j}| ≥ T/R, we get from Hölder’s inequlity:

∑_{|λ_{j}|≥ \frac{T}{R}} |λ_{j}|^{\frac{1}{p}} ≤ N^{\frac{1}{q}} ( ∑_{|λ_{j}|≥ \frac{T}{R}} |λ_{j}|)^{\frac{1}{p}} .  (5.35)

Here NT/R ≤ ∑ |λ_{j}| ≤ C, so N ≤ CR/T and the expression (5.35) is bounded by CR^{1/q}/T^{1/q}. Hence

∑_{|λ_{j}|R≥T} \frac{1}{R} ∫_{|w|=R} | log |1 − λ_{j}w||ℓ(dw) ≤ CR^{\frac{1}{p} + \frac{1}{q}} /T^{\frac{1}{q}} = \frac{CR}{T^{\frac{1}{q}}}.  (5.36)

Combining the three cases, we find

| ∫_{|w|=R} P_{R}(z, w)(∑_{1}^{∞} log |1 − λ_{j}w|)ℓ(dw)| ≤ o_{T} (R) + \frac{CR}{T^{\frac{1}{q}}} = o(R), R → ∞

(5.37)

Combining this with (5.28), (5.29) and (5.31), we get

Re g(z) ≤ o(R), on |z| ≤ \frac{R}{2}.

Now we can apply Harnack’s inequality to the function Re g − o(R), which is ≤ 0 on the disc |z| ≤ R/2 and ≥ −o(R) at 0 and conclude that

Re g ≥ −o(R) on the disc |z| ≤ \frac{R}{4}.

Since g is harmonic, it follows from the last two estimates that Re g = 0. Hence g is constant and since we have chosen g with g(0) = 0, we get g(z) = 0 ∀z ∈ C. We have then showed

Theorem 5.14 Let A : H → H be a trace class operator with the nonvanishing eigenvalues λ_{1}(A), λ_{2}(A), .., 0 ≤ N ≤ ∞ repeated according to multiplicity. Then D_{A}(z) = det(1 − zA) satisfies

D_{A}(z) = ∏_{j=1}^{N}(1 − λ_{j}z), z ∈ C.  (5.38)

From this we get the important Lidskii’s theorem as a corollary :

Corollary 5.15 If A ∈ C_{1} we have

tr A = ∑_{1}^{N} λ_{j}(A),  (5.39)

where λ_{j}(A) are the non-vanishing eigenvalues as in Theorem 5.14.

Proof. We know the result when A is of finite rank. In this case, we also know that

\frac{D′_{A}}{D_{A}} = \frac{∂}{∂z} log det(1 − zA) = −tr ((1 − zA)^{−1}A),

away from the zeros of z ↦ det(1 − zA). In particular,

\frac{D′_{A}(0)}{D_{A}(0)} = −tr A,  (5.40)

when A is of finite rank.

When A ∈ C_{1}, let A_{ν} be a sequence of finite rank operators converging to A in the C_{1} norm. Then D_{A_{ν}}(z) → D_{A}(z), D′_{A_{ν}}(z) → D′_{A}(z), when ν → ∞, uniformly for z in a neighborhood of 0. Since D_{A}(z) ≠ 0 in such a neighborhood, we have also

\frac{D′_{A_{ν}}(0)}{D_{A_{ν}}(0)} → \frac{D′_{A}(0)}{D_{A}(0)}.

By (5.40) we know that the right hand side of the last relation is equal to −tr A_{ν} and we also know that this quantity converges to −tr A. Consequently (5.40) remains valid for general trace class operators. In view of Theorem 5.14, we know on the other hand that

\frac{D′_{A}(0)}{D_{A}(0)} = − ∑_{1}^{N} λ_{j}(A),

and the Corollary follows. #

The last proof also shows that

\frac{D′_{A}(z)}{D_{A}(z)} = \frac{∂}{∂z} log D_{A}(z) = −tr (1 − zA)^{−1}A = − ∑_{1}^{N} \frac{λ_{j}(A)}{1 − zλ_{j}(A)}, (5.41)

for all z with 1 − zλ_{j}(A) ≠ 0, ∀j.

We now turn to some other questions, that will be of use.

Proposition 5.16 Let H_{0}, H_{1}, H_{2} be complex separable Hilbert spaces. Let Ω ⊂ C^{n} be an open set and let Ω ∋ z ↦ K(z) ∈ C_{1}(H_{1}) be a holomorphic function. Then det(1 − K(z)) is a holomorphic function on Ω and if z_{0} ∈ Ω is a zero of order m ≥ 1 of this function, and A(z) ∈ L(H_{1}, H_{2}), B(z) ∈ L(H_{0}, H_{1}) depend holomorphically on z ∈ Ω, then

rank (∫_{γ} A(z)(1 − K(z))^{−1}B(z)dz) ≤ m,  (5.42)

if γ is the positively oriented boundary of a sufficiently small circle centered at z_{0}.

Proof. We know that 1 is an eigenvalue of K(z_{0}) of a certain multiplicity N_{0}, defined to be the rank of the spectral projection

Π(z_{0}) = \frac{1}{2πi} ∫_{α}(λ − K(z_{0}))^{−1}dλ,

where α is the positively oriented boundary of a small disc centered at λ = 1. For z close to z_{0}, we put

Π(z) = \frac{1}{2πi} ∫_{α}(λ − K(z))^{−1}dλ,

and we notice that this is the sum of the spectral projections corresponding to the N_{0} eigenvalues λ_{j_{0}}(z), λ_{j_{0}+1}(z), .., λ_{j_{0}+N_{0}−1}(z) (repeated according to multiplicity) that are close to λ_{j_{0}}(z_{0}). m is then also the order of vanishing of (1 − λ_{j_{0}}(z)) · .. · (1 − λ_{j_{0}+N_{0}−1}(z)) (if we note that det(1 − K(z)) = det(1 − K(z)Π(z)) det(1 − K(z)(1 − Π(z))) ). The range of Π(z) is of constant dimension N_{0} and we can find a basis e_{1}(z), .., e_{N_{0}}(z) of this space which depends holomorphically on z (possibly after restricting z to a new even smaller neighborhood of z_{0}.

Define R_{+} : H_{1} → C^{N_{0}} by R_{+}(z)u(j) = a_{j}(u, z), where Π(z)u = ∑_{1}^{N_{0}} a_{j}(u, z)e_{j}. Define R_{−}(z) : C^{N_{0}} → H_{1} by R_{−}(z)u_{−} = ∑_{1}^{N_{0}} u_{−}(j)e_{j}(z). Then

( \begin{matrix}1 − K(z) & R_{−}(z) \\ R_{+}(z) & 0 \end{matrix} ) : H_{1} × C^{N_{0}} → H_{1} × C^{N_{0}}

is bijective with inverse

( \begin{matrix}E(z) & E_{+}(z) \\ E_{−}(z) & E^{−+}(z) \end{matrix} )

where E_{−+} is the matrix of the restriction of K(z)−1 to R(Π(z)) with respect to the basis e_{1}(z), .., e_{N_{0}}(z). Hence det E_{−+}(z) = ∏_{ν=0}^{N_{0}−1}(1 − λ_{j_{0}}(z))...(1 − λ_{j_{0}+N_{0}−1}(z)) has the same order of vanishing at z = z_{0} as det(1 − K(z)). It then suffices to apply Lemma [[... som skall ingaa i en foersta del av detta kapitel daer vi behandlar Fredholm teori med hjaelp av Grushin problem]].

We end this chapter by recalling Jensen’s formula and the standard application to getting bounds on the number of zeros of holomorphic functions. Let f(z) be a holomorphic function on the open disc D(0, R) with a continuous extension to the corresponding closed disc. Assume that f(0) ≠ 0 and f(z) ≠ 0 for |z| = R.

Assume first that f(z) has no zeros at all. Then log |f(z)| = Re log f(z) is a harmonic function in the open disc which is continuous up to the boundary, and the mean value property of harmonic functions tells us that

log |f(0)| = \frac{1}{2π} ∫_{0}^{2π} log |f(Re^{iθ})|dθ.

We now allow f to vanish and let z_{1}, .., z_{N} be the zeros repeated according to their multiplicity. Since f is not allowed to vanish at 0 or at the boundary of the disc of radius R, we have 0 < |z_{j}| < R. Then

F (z) := f(z) ∏_{j=1}^{N} \frac{R^{2} − \overline{z_{j}}z}{R(z − z_{j})}

is holomorphic in the open disc, continuous up to the boundary and has no zeros in the closed disc of radius R. Moreover |F (z)| = |f(z)| when |z| = R, so according to the preceding paragraph, we have

log |F (0)| = \frac{1}{2π} ∫_{0}^{2π} log |f(Re^{iθ})|dθ.

Expanding the left hand side, we get Jensen’s formula :

log |f(0)| + ∑_{1}^{N} log \frac{R}{|z_{j}|} = \frac{1}{2π} ∫_{0}^{2π} log |f(Re^{iθ})|dθ.  (5.43)

A standard application of this formula is to notice that if N(R/2) is the number of zeros z_{j} of f with |z_{j}| ≤ R/2, then we get

N(\frac{R}{2}) log 2 ≤ \frac{1}{2π} ∫_{0}^{2π} log |f(Re^{iθ})|dθ − log |f(0)|.  (5.44)

6 Global upper bounds on the number of resonances

In this chapter we shall show that in standard situations and for odd dimensions ≥ 3, the number of resonances in a disc D(0, r) (of center 0 and radius r) is O(r^{n}), r → ∞. It will be convenient to formulate the result in the black box setting of section 2.3 of chapter 2, and after that we will mention some of the history of this result. The methods come to a large extent from the theory of non-selfadjoint operators, and can therfore be viewed as classical.

Let P : H → H be as in chapter 2, section 2.3, so that H is given by the orthogonal decomposition (2.30). Introduce the torus M = (R/2RZ)^{n} with R > R_{0} and identify \overline{B(0, R_{0})} with its image under the natural projection R^{n} → M. Put H^{♯} → H^{♯}. The domain D^{♯} = D(P^{♯}) is by definition

D^{♯} = {χu_{1} + (1 − χ)u_{2}; u_{1} ∈ D, u_{2} ∈ H^{2}(M)}.  (6.1)

Here χ ∈ C_{0}^{∞} has its support in a small neighborhood of \overline{B(0, R_{0})} and is equal to one near that set. We view χ both as an element of C_{0}^{∞}(R^{n}) and as an element of C_{0}^{∞}(M). It is easy to see that the definition (6.1) does not depend on the choice of χ. For D(P^{♯}) ∋ u = χu_{1} + (1 − χ)u_{2}, we put

P^{♯}u = P (χu_{1}) + (−Δ)((1 − χ)u_{2}),  (6.2)

and again, this does not depend on the representation of u.

Lemma 6.1 P^{♯} is self-adjoint and has purely discrete spectrum.

Proof. It is easy to see that P^{♯} is symmetric and we leave the details to the reader. Let u ∈ D((P^{♯})^{∗}), (P^{♯})^{∗}u = v, v ∈ H^{♯}. We claim that u ∈ D(P^{♯}): First we use that

(u| − Δϕ) = (v|ϕ), ∀ϕ ∈ C_{0}^{∞}(M \ \overline{B(0, R_{0})}),

so that −Δu = v in M \ \overline{B(0, R_{0})} and hence

u_{|}_{M\\overline{B(0,R_{0})}} ∈ H_{loc}^{2}(M \ \overline{B(0, R_{0})}).

After subtracting an exterior part from u, we may assume that χu = u, with χ as above. Now we can view u, v as elements of H, and from the relation (P^{♯})^{∗}u = v, it follows that

(u|P ϕ) = (v|ϕ),

first for all ϕ ∈ D with support in a neighborhodd of supp χ, then for all ϕ ∈ D. Since P is self-adjoint, we get u ∈ D and the claim follows.

Let v_{j} ∈ H^{♯}, j = 1, 2, ... be a bounded sequence and consider u_{j} = (P^{♯} + i)^{−1}v_{j} ∈ D^{♯}. With χ as before, we first see that (1−χ)(P^{♯}+i)^{−1}v_{j} = (1−χ)u_{j} is bounded in H^{2}(M) and hence has a convergent subsequence in H^{♯}. As for χu_{j}, we compute (P + i)χu_{j} = [−Δ, χ]u_{j} + χv_{j} =: w_{j}, so w_{j} is bounded in H. Now write χu_{j} = (P + i)^{−1}w_{j} = (P + i)^{−1}χ̃w_{j}, where χ ≺ χ̃ ∈ C_{0}^{∞}(R^{n}). Since (P + i)^{−1}χ̃ is compact, χu_{j} has a convergent subsequence in H and in H^{♯}. It follows that (P^{♯} + i)^{−1} : H^{♯} → H^{♯} is compact. Hence P^{♯} has purely discrete spectrum. #

Let N(P^{♯}, I) be the number of eigenvalues of P^{♯} in the interval I. We assume

N(P^{♯}, [−λ, λ]) = O(1)Φ(λ), λ ≥ 1,  (6.3)

where Φ : [1, ∞[→ [1, ∞[ is continuous strictly increasing, with Φ(1) = 1. We also assume:

Φ(t) ≥ t^{\frac{n}{2}} ,  (6.4)
∀C ≥ 1, ∃C̃(C) ≥ 1, such that  (6.5)
Φ(Ct) ≤ C̃(C)Φ(t), ϕ(Cλ) ≤ C̃(C)ϕ(λ), λ, t ≥ 1,

where ϕ(λ) = Φ^{−1}(λ).

Let λ_{1}, λ_{2}, λ_{3}, ... be the eigenvalues of P^{♯} repeated according to multiplicity and arranged so that |λ_{1}| ≤ |λ_{2}| ≤ .... The eigenvalues of the compact normal operator (i−P^{♯})^{−1} are then (i−λ_{j})^{−1}, and the corresponding singular values are given by

s_{j}((i − P^{♯})^{−1}) = \frac{1}{|i − λ_{j}|} = \frac{1}{⟨λ_{j}⟩}.  (6.6)

It is easy to see that the property (6.3) is equivalent to

s_{j}((i − P^{♯})^{−1}) ≤ \frac{O(1)}{ϕ(j)}.  (6.7)

For the same operator P, let M̃ be a second torus and P̃ the corresponding operator analogous to P^{♯}. We identify M and M̃ by means of a diffeomorphism which acts like the identity near \overline{B(0, R_{0})}. In this way, we can view both P and P^{♯} as operators on M (i.e. as operators H^{♯} → H^{♯}.

(Notice however, that P̃ will not be self-adjoint with respect to the scalar product of H^{♯} but for another one which gives an equivalent norm.) We have the resolvent identity:

(i − P̃)^{−1} = (i − P^{♯})^{−1} + (i − P̃)^{−1}(P̃ − P^{♯})(i − P^{♯})^{−1}.  (6.8)

Now recall some facts from Chapter 5 about singular values. If K : H → H is a compact operator, then the jth singular value (i.e. the jth eigenvalue of \sqrt{K^{∗}K} counted with multiplicities in decreasing order) obeys Ky Fan’s identity

s_{j}(K) = inf_{{R∈L(H,H), \atop rank (R)≤j−1}} ∥K − R∥.  (6.9)

From this we deduce the wellknown inequalities:

s_{j+k−1}(K + L) ≤ s_{j}(K) + s_{k}(L),  (6.10)
s_{j+k−1}(KL) ≤ s_{j}(K)s_{k}(L),  (6.11)

for j, k ≥ 1. Applying this to (6.8), we see that P̃ also satisfies (6.7). This shows that the assumption (6.3) does not depend on the choice of M.

Later on we will also need the following information which follows from (6.9): If K is a compact operator, then for every j ≥ 1, there exists a bounded operator R of rank ≤ j − 1, such that

s_{ν}(K) = s_{ν}(R), 1 ≤ ν ≤ j − 1,  (6.12)
s_{j}(K) = ∥K − R∥ = s_{1}(K − R),
s_{ν}(K) = s_{ν−j+1}(K − R), ν ≥ j.

The same argument shows that

s_{j}((P − z)^{−1}χ), s_{j}(χ(P − z)^{−1}) ≤ \frac{C}{ϕ(j)},  (6.13)

for χ ∈ C_{0}^{∞}(R^{n}), 1_{B(0,R_{0})} ≺ χ and for z in any fixed comapct subset of the resolvent set. This estimate is the starting point of the proof of the following result:

Theorem 6.2 Let n ≥ 3 be odd. Under the assumptions above, the number N(r) of resonances in the open disc D(0, r) satisfies

N(r) ≤ CΦ(r^{2}), r ≥ 1.  (6.14)

Before starting the proof we give some examples and make some historical remarks. Let g be a smooth Riemannian metric on R^{n} which coincides with the standard Euclidean metric on R^{n} outside a bounded set, let O ⊂⊂ R^{n} be open with smooth boundary, let V ∈ L_{comp}^{∞}(R^{n} \ O), and let P be the Dirichlet or Neumann realization in L^{2}(R^{n} \ O) of the Schrödinger operator −Δ_{g} + V (x), where Δ_{g} denotes the Laplace-Beltrami operator for the metric g. Then we can take Φ(r) = r^{n/2} and we get N(r) ≤ Cr^{n}. In the case, V = 0 and Δ_{g} = Δ everywhere, this was proved by Melrose [54]. Zworski [101] obtained it for −Δ + V without any obstacle, and Vodev [92] obtained it for −Δ_{g} on L^{2}(R^{n}). Theorem 6.2 as stated above is essentially due to Sjöstrand–Zworski [80], but in the proof below we shall mainly follow Vodev [93], who extended the result to certain non-selfadjoint operators and who also obtained sharp results in the even-dimensional case [95, 96]. In the context of hyperbolic manifolds, analogous results have been obtained by Guillopé–Zworski [34], Froese–Hislop [25].

Proof. Let

1_{B(0,R_{0})} ≺ χ_{0} ≺ χ_{1} ≺ χ_{2}, χ_{j} ∈ C_{0}^{∞}(R^{n})

as in Chapter 2. Let μ ∈ C, arg μ = \frac{π}{4}, |μ| = r ≫ 1. Then we know that the resonances λ are contained in the set of λ, such that (1+K(λ, μ)χ) : H → H is not invertible, where χ_{2} ≺ χ ∈ C_{0}^{∞}(R^{n}). Here K is the same as in Chapter 2, (2.39), (2.40), (2.41):

K(λ, μ) = [Δ, χ_{0}]R_{0}(λ)(1−χ_{1})−[Δ, χ_{2}]R(μ)χ_{1} +(μ^{2} −λ^{2})χ_{2}R(μ)χ_{1}. (6.15)

We will restrict the attention to {λ ∈ C; |λ − μ| < 3r}. Since K(λ, μ)χ is not necessarily of trace class, we shall first decompose it into one term with small norm, and one term which is of trace class. This decomposition only concerns the last term in (6.15).

Write

R(μ)χ_{1} = (P − i|μ|^{2})^{−1}(P − i)(P − i)^{−1}χ,

and conclude that

s_{j}(χ_{2}R(μ)χ_{1}) ≤ min(\frac{1}{r^{2}} , \frac{O(1)}{ϕ(j)}), j = 1, 2, ...  (6.16)

For j(r) = O(1)Φ(r^{2}), we can decompose :

χ_{2}R(μ)χ_{1} = A_{μ} + B_{μ},  (6.17)

where

∥A∥ ≤ \frac{1}{100r^{2}} ,  (6.18)
s_{j}(B) = \{ \begin{matrix}s_{j}(χ_{2}R(μ)χ_{1}), 1 ≤ j ≤ j(r), \\ 0, j ≥ j(r) + 1. \end{matrix}  (6.19)

We have already seen in Chapter 2, that if |μ| = r is large enough, then

∥[Δ, χ_{2}]R(μ)χ_{1}∥ ≤ \frac{1}{4}.  (6.20)

Write

1 + K(λ, μ) =  (6.21)
1 + D + (μ^{2} − λ^{2})B + [Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ,

where D = (μ^{2} − λ^{2})A − [Δ, χ_{2}]R(μ)χ_{1} and ∥D∥ ≤ 1/2,

1 + K(λ, μ)χ = (1 + D)(1 + K̃),  (6.22)
K̃ = (μ^{2} − λ^{2})(1 + D)^{−1}B + (1 + D)^{−1}[Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ.

We shall take the determinant of 1 + K̃, and in order to estimate this determinant, we shall first estimate the singular values of the last term in the expression for K̃.

For Im λ ≥ 0, α > 0, we have

s_{j}([Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ) ≤ C_{α}⟨λ⟩^{α}j^{−α/n},  (6.23)

and for Im λ < 0 :

s_{j}([Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ) ≤ \{ \begin{matrix}e^{C⟨λ⟩}, ∀j \\ j^{− \frac{n}{n−1}} + C_{α}⟨λ⟩^{α}j^{\frac{−α}{n}} , j ≥ C⟨λ⟩^{n−1}. \end{matrix} (6.24)

We postpone the proof of these two estimates and finish the proof of the theorem. Let

h(λ, μ) = det(1 + K̃)

For λ = μ, we get

h(μ, μ) = det(1 + (1 + D)^{−1}[Δ, χ_{0}]R_{0}(μ)(1 − χ_{1})χ).

In this case we can use the off diagonal exponential decay of the distribution kernel of R_{0}(μ), to see that the distribution kernel k(x, y) of [Δ, χ_{0}]R_{0}(μ)(1− χ_{1})χ satsifies:

∑_{|α|≤2n+1} ∥∂_{x,y}^{α}k(x, y)∥_{L^{1}(R^{2n})} ≤ O(1)e^{−r/C_{0}},  (6.25)

for some C_{0} > 0. On the other hand it is well known that the trace class norm of [Δ, χ_{0}]R_{0}(μ)(1 − χ_{1})χ is bounded by a constant times the left hand side of (6.25), so

∥(1 + D)^{−1}[Δ, χ_{0}]R_{0}(μ)(1 − χ_{1})χ∥_{tr} ≤ O(1)e^{−r/C_{0}}.  (6.26)

Use also the Weyl inequality

| det(1 + K̃)| ≤ ∏_{1}^{∞}(1 + s_{j}(K̃)) ≤ e^{∥_{K}∥_{tr}},  (6.27)

to conclude that

\frac{1}{C} ≤ |h(μ, μ)| ≤ C,  (6.28)

for r = |μ| sufficiently large.

We shall next find upper bounds for |h(λ, μ)|. It follows from (6.23), (6.24), that (6.24) holds also in the upper half plane. We may assume that ∥(1 + D)^{−1}∥ ≤ 2 and get

|h(λ, μ)| ≤ ∏_{j=1}^{∞}(1 + 2s_{j}((μ^{2} − λ^{2})B + [Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ)).

Use that for j = 2k − 1: s_{j}(A_{1} + A_{2}) ≤ s_{k}(A_{1}) + s_{k}(A_{2}) by Corollary 5.2:

|h(λ, μ)| ≤ ∏_{k=1}^{∞}(1 + 2s_{2k−1}((μ^{2} − λ^{2})B) + [Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ))^{2}

≤ ∏_{k=1}^{∞}(1 + 2s_{k}((μ^{2} − λ^{2})B))^{2} ∏_{k=1}^{∞}(1 + 2s_{k}([Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ))^{2}

= F_{1}F_{2},

where the last equation defines the two factors F_{j} in the natural way. Clearly, F_{1} ≤ exp(O(1)r^{2} ∑_{1}^{∞} s_{k}(B)). Hence by (6.16), (6.19):

∑_{1}^{∞} s_{k}(B) ≤ ∑_{1}^{j(r)} min(\frac{1}{r^{2}} , \frac{O(1)}{ϕ(k)}) ≤ ∑_{1}^{O(1)Φ(r^{2})} \frac{1}{r^{2}} ≤ O(1)\frac{Φ(r^{2})}{r^{2}}.

Hence

F_{1} ≤ exp(C_{1}Φ(r^{2})).  (6.29)

Using (6.24):

F_{2} ≤ ( ∏_{1≤k≤C⟨λ⟩^{n−1}} e^{2C⟨λ⟩}) ×

exp(4 ∑_{k>C⟨λ⟩^{n−1}} j^{− \frac{n}{n−1}} + C_{α_{1}} ∑_{C⟨λ⟩^{n−1}≤j≤⟨λ⟩^{n}}⟨λ⟩^{α_{1}}j^{− \frac{α_{1}}{n}} + C_{α_{2}} ∑_{j>⟨λ⟩^{n}}⟨λ⟩^{α_{2}}j^{− \frac{α_{2}}{n}}),

where we are free to choose α_{1}, α_{2} > 0. Take α_{1} < n < α_{2}. Then evaluating the three sums in the last exponent, we get

F_{2} ≤ exp C⟨λ⟩^{n}.  (6.30)

Consequently, for |λ − μ| ≤ 3r, arg μ = \frac{π}{4}, |μ| = r:

|h(λ, μ)| ≤ e^{C(r^{n}+Φ(r^{2}))} ≤ e^{_{C}Φ(r^{2})}.  (6.31)

by Jensen’s formula for the disc D(μ, 3r), we see that the number of zeros of λ → h(λ, μ) in D(μ, 2r) is ≤ O(1)Φ(r^{2}). Now write (cf (2.46))

R(λ) = Q(λ, μ)(1 + K(λ, μ)χ)^{−1}(1 − K(λ, μ)(1 − χ))
= Q(λ, μ)(1 + K̃(λ, μ))^{−1}(1 + D)^{−1}(1 − K(λ, μ)(1 − χ))
= A(λ, μ)(1 + K̃(λ, μ))^{−1}B(λ, μ),

where A, B are holomorphic in λ. After multiplying to the left and to the right by functions in C_{0}^{∞}, we can apply the propositions 5.16, 2.3, and conclude that the number of resonances in D(μ, 2r) is bounded by the number of zeros of h(·, μ) in that set and hence by O(1)Φ(r^{2}). The theorem follows. #

It remains to prove (6.23), (6.24). We start with the first of these estimates. For |λ| ≤ 1, we use that [Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ has a distribution kernel in C_{0}^{∞}. If Ω is a large ball, we can consider our operator on Ω and write

[Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ = (1 − Δ_{Ω})^{−N}(1 − Δ_{Ω})^{N}[Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ,

where Δ_{Ω} denotes the Dirichlet Laplacian in Ω. Here

(1 − Δ_{Ω})^{N}[Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ

is uniformly bounded and the jth singular value of (1 − Δ_{Ω})^{−N} is ∼ j^{− \frac{2}{n} N} by the known (Weyl) asymptotics for the eigenvalues of −Δ_{Ω}.

For |λ| ≥ 1, we follow the same idea, but we have to be more precise. Let us first show that if χ_{1}, χ_{2} ∈ C_{0}^{∞}, then

χ_{1}R_{0}(λ)χ_{2} = O_{s}(\frac{1}{|λ|}) : H_{s} → H_{s}, s ∈ R, Im λ ≥ 0.  (6.32)

Indeed, on the Fourier transform side, the kernel of χ_{1}R_{0}(λ)χ_{2} becomes

K(ξ, η) = ∫ χ̂_{1}(ξ − ζ)\frac{1}{ζ^{2} − λ^{2}} χ̂_{2}(ζ − η)\frac{dζ}{(2π)^{n}} , ξ, η ∈ R^{n}.

If |ζ^{2} − λ^{2}| ≥ \frac{|λ|}{C}, ζ ∈ R^{n}, then |K(ξ, η)| ≤ C_{N} \frac{⟨ξ−η⟩^{−N}}{|λ|} and χ_{1}R(λ)χ_{2} = O_{s}(\frac{1}{|λ|}) : H^{s} → H^{s}, ∀s.

If |ζ^{2}−λ^{2}| is smaller than \frac{|λ|}{C}, then |ζ| ∼ |λ|, and hence |∇_{ζ}(ζ^{2}−λ^{2})| ∼ |λ|. We then replace R_{ζ}^{n} by the deformed contour:

Γ : R^{n} ∋ ζ ↦ ζ ± iϵχ(\frac{|ζ|^{2} − |λ|^{2}}{|λ|^{2}} )\frac{ζ}{|ζ|} = ζ + iO(ϵ) =: ζ̃,

for a suitable χ, and along Γ, we have

ζ̃^{2} = ζ^{2} ± 2iϵχ(...)\frac{ζ^{2}}{|ζ|} − ϵ^{2}χ^{2}(...),

and we get |ζ̃^{2} − λ^{2}| ≥ |λ|/C. Using Paley–Wiener, we obtain (6.32). Next we show that if χ_{1}, χ_{2} ∈ C_{0}^{∞}(R^{n}), supp χ_{1} ∩ supp χ_{2} = ∅, then

χ_{1}R_{0}(λ)χ_{2} = O_{s,m}(⟨λ⟩^{m−1}) : H^{s} → H^{s+m}, for m ≥ 0.  (6.33)

In fact, if m = 2k, we notice that

(−Δ + 1)R_{0}(λ) = (1 + λ^{2})R_{0}(λ) + 1,

and hence outside supp χ_{2}:

(−Δ + 1)R_{0}(λ)χ_{2} = (1 + λ^{2})R_{0}(λ)χ_{2}.

By iteration, we get toutside supp χ_{2}:

(−Δ + 1)^{k}R_{0}(λ)χ_{2} = (1 + λ^{2})^{k}R_{0}(λ)χ_{2},

so

χ_{1}(−Δ + 1)^{k}R_{0}(λ)χ_{2} = (1 + λ^{2})^{k}χ_{1}R_{0}(λ)χ_{2}.

This gives (6.33) for m = 2k, and by interpolation, we get it for general m ≥ 0. Proof of (6.23). Let Ω be as before, so that the jth eigenvalue of (−Δ_{Ω} + 1)^{1/2} is ∼ j^{1/n}. Then as before,

[Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ = (1 − Δ_{Ω})^{− \frac{α}{2}} \underbrace{(1 − Δ_{Ω})^{\frac{α}{2}} [Δ, χ_{0}]R_{0}(λ)(1 − χ_{1})χ}_{=O(⟨λ⟩^{α}) in operator norm} .

It suffices to use s_{j}((1 − Δ_{Ω})^{− \frac{α}{2}} ) ∼ j^{− \frac{α}{n}} # Proof of (6.24). Thanks to (2.5), (6.23) it suffices to show that

s_{j}(q_{1}T (λ)q_{0}) ≤ \{ \begin{matrix}e^{C⟨λ⟩}, ∀j, \\ j^{− \frac{n}{n−1}} , j ≥ C⟨λ⟩^{n−1}, \end{matrix}  (6.34)

where q_{1}, q_{0} are differential operators with C_{0}^{∞} coefficients of order ≤ 1 and 0 respectively and where T (λ) is given by (2.6). Clearly, ∥T (λ)∥ ≤ e^{C⟨λ⟩} so the first estimate in (6.34) is obvious.

We observe that T (λ) = λ^{n−2}E(\overline{λ})^{∗}E(λ), where

E(λ)u(ω) = ∫ e^{−iλω·y}χ(y)u(y)dy.  (6.35)

Write

T (λ) = λ^{n−2}E(\overline{λ})^{∗}(1 − Δ_{S^{n−1}})^{−k}(1 − Δ_{S^{n−1}})^{k}E(λ).  (6.36)

Here

s_{j}((1 − Δ_{S^{n−1}})^{−k}) ≤ (\frac{j}{C} )^{− \frac{2k}{n−1}} ,  (6.37)
∥(1 − Δ_{S^{n−1}})^{k}E(λ)q_{0}∥ ≤ e^{2|λ|}(Ck)^{2k},  (6.38)

where the last estimate follows from the Cauchy inequalities. We deduce that

s_{j}(q_{1}T (λ)q_{0}) ≤ Ce^{3|λ|}(\frac{C}{j} )^{\frac{2k}{n−1}} (Ck)^{2k}  (6.39)

≤ Ce^{3|λ|}(\frac{C̃k}{j^{1/(n−1)}} )^{2k}.

Choose k so that

\frac{C̃k}{j^{1/(n−1)}} = \frac{1}{e},

i.e. so that

k = \frac{j^{1/(n−1)}}{C̃e} .

Actually we have to modify this choice a little bit since k should be an integer.) Then (6.39) gives

s_{j}(T (λ)) ≤ C^{3|λ|}e^{−2 \frac{j \frac{1}{n−1}}{_{C}e}} .  (6.40)

For j^{\frac{1}{n−1}} ≫ |λ|, we get

s_{j}(T (λ)) ≤ Ce^{− \frac{j \frac{1}{n−1}}{_{C}e}} ,

which is even better than (6.34). This completes the proof of (6.24). #

7 Resonances for long range perturbations of −h^{2}Δ, analytic distorsions

7.1 Operators of black box type

Our operators will depend on a Planck’s constant h ∈]0, h_{0}], h_{0} > 0. Let H be a complex separable Hilbert space with an orthogonal decomposition

H = H_{R_{0}} ⊕ L^{2}(R^{n} \ B(0, R_{0})), B(0, R_{0}) = {x ∈ R^{n}; |x| < R_{0}}, R_{0} > 0.

(7.1)

The orthogonal projections will be denoted by u ↦ u_{|}_{B(0,R_{0})}, u ↦ u_{|}_{R^{n}\B(0,R_{0})}. Sometimes we let the characteristic functions 1_{B(0,R_{0})}, 1_{R^{n}\B(0,R_{0})} indicate the same projections.

Consider an unbounded self-adjoint operator depending on h:

P : H → H  (7.2)

with domain D = D(P). Recall that D is a Hilbert space with norm ∥u∥_{D} = ∥(P + i)u∥_{H}. Let H^{s}(R^{n}) be the standard Sobolev space, but equipped with the h-dependent norm ∥u∥_{H^{s}} = ∥⟨hD⟩^{s}u∥_{L^{2}}. Assume that for every 1_{B(0,R_{0})} ≺ χ ∈ C_{0}^{∞}(R^{n}) independent of h,

the multiplication operators 1 − χ : L^{2}(R^{n}) → H, H → L^{2}(R^{n}) (7.3)

restrict to bounded operators H^{2}(R^{n}) → D, D → H^{2}(R^{n}), with norms bounded uniformly in h.

Assume

χ(P + i)^{−1} is compact,  (7.4)

for every χ as in (7.3). This assumption will be strengthened later. Further assume that

(P u)_{|}_{R^{n}\B(0,R_{0})} = Q(u_{|}_{R^{n}\B(0,R_{0})}), u ∈ D,  (7.5)

where

Qu = ∑_{|α|≤2} a_{α}(x; h)(hD_{x})^{α}u,  (7.6)
a_{α}(x; h) = a_{α}(x) is independent of h for |α| = 2,
and a_{α} is bounded in C_{b}^{∞}(R^{n}) when h varies .

We further assume that Q is formally self-adjoint on R^{n} with

∑_{|α|=2}a_{α}(x)ξ^{α} ≥ \frac{1}{C} |ξ|^{2},  (7.7)
∑_{|α|≤2}a_{α}(x; h)ξ^{α} → ξ^{2}, |x| → ∞, uniformly w.r.t. h.  (7.8)

Example 1. Let O ⊂⊂ R^{n} be an open set with smooth boundary and let P be the Dirichlet or Neumann realization of −Δ in L^{2}(R^{n} \ O). Then (cf. Chapter 2), h^{2}P satisfies the assumptions above, with H = L^{2}(R^{n} \ O). (Take R_{0} large enough so that the ball of radius R_{0} contains O.) Example 2. P = −h^{2}Δ + V (x), where V ∈ C_{b}^{∞}(R^{n}) with V (x) → 0, x → ∞. H = L^{2}(R^{n}).

Let R > R_{0}, T = (R/R̃Z)^{n}, R̃ > 2R. We can view B(0, R) as a subset of T. Put

H^{♯} := H_{R_{0}} ⊕ L^{2}(T \ B(0, R_{0})),  (7.9)

with orthogonal decomposition. We define a self-adjoint operator

P^{♯} : H^{♯} → H^{♯}

as follows: The domain D^{♯} = D(P^{♯}) is by definition

{u ∈ H^{♯}; χu ∈ D, (1 − χ)u ∈ H^{2}(T)}

where 1_{B(0,R_{0})} ≺ χ ∈ C_{0}^{∞}(B(0, R); [0, 1]). (This does not depend on the choice of χ.) Let

Q^{♯}u = ∑_{|α|≤2} a_{α}^{♯}(x; h)(hD)^{α}

be a formally selfadjoint differential operator on T satisfying (7.6), (7.7) and with a_{α}^{♯}(x; h) = a_{α}(x; h) for |x| < R. With χ as in the definition of D^{♯}, we put

P^{♯}u = P χu + Q^{♯}(1 − χ)u, u ∈ D^{♯}.  (7.10)

We have

Proposition 7.1 P^{♯} is self-adjoint with discrete spectrum.

Proof. It is easy to see that P^{♯} is symmetric. Let u ∈ D((P^{♯})^{∗}), and write (P^{♯})^{∗}u = v, with v ∈ H^{♯}. Then

(u|Q^{♯}ϕ) = (v|ϕ),

for all u ∈ C_{0}^{∞}(T \ \overline{B(0, R_{0})}), so Q^{♯}u = v in T \ B(0, R_{0}), in the sense of distributions. From the ellipticity of the operator in this region, it follows that

u_{|}_{T \\overline{B(0,R_{0})}} ∈ H_{loc}^{2}(T \ \overline{B(0, R_{0})}).

After subtracting an exterior part from u, of class H^{2} and hence belonging to D(P^{♯}), we may assume that χu = u, with χ as in the definition of D(P^{♯}).

We now have

(u|P ϕ) = (v|ϕ),  (7.11)

for all ϕ ∈ D with support in B(0, 2R). (Here we may view u, v as elements of H since their supports are contained in B(0, R).) Because of the support properties of u, v, we see that (7.11) extends to general ϕ ∈ D and the selfadjointness of P implies that u ∈ D, and hence u ∈ D^{♯} by the support property of u. This finishes the proof of the self-adjointness of P^{♯}.

To prove that P^{♯} has discrete spectrum, we shall use the assumption (7.4). Let v_{j} ∈ H^{♯}, j = 1, 2, .. be a bounded sequence and put u_{j} = (P^{♯} + i)^{−1}v_{j} ∈ D^{♯}. With χ as above, we notice that (1 − χ)(P^{♯} + i)^{−1}v_{j} = (1 − χ)u_{j} is a bounded sequence in H^{2}(T) and hence has a convergent subsequence in H^{♯}. As for χu_{j}, we compute

(P + i)χu_{j} = \underbrace{[Q, χ]u_{j}}_{bounded in H} + \underbrace{χv_{j}}_{bounded in H} =: w_{j},

so χu_{j} = (P + i)^{−1}w_{j} = (P + i)^{−1}χ̃w_{j}, if χ ≺ χ̃ ∈ C_{0}^{∞}. Now we notice that by the resolvent identity, (7.4) implies the compactness of χ(P − z)^{−1} for every z ∈ C \ σ(P). Here we can replace χ by χ̃ and the compactness of χ̃(P − i)^{−1} implies that of (P + i)^{−1}χ̃ by passing to adjoints. Hence χu_{j} also has a convergent subsequence. It follows that (P^{♯} + i)^{−1} is compact or equivalently, that P^{♯} has a purely discrete spectrum. #

Later on we shall need assumptions on the density of eigenvalues of P^{♯}. Let N(P^{♯}, I) denote the number of eigenvalues of P^{♯} in the interval I. We assume

N(P^{♯}, [−λ, λ]) = O(1)Φ(\frac{λ}{h^{2}} ), λ ≥ 1.  (7.12)

Here Φ : [1, ∞[→ [1, ∞[ is a continuous strictly increasing function with Φ(t) ≥ t^{n/2} such that:

For every C ≥ 1, there exists C̃(C) > 0, such that Φ(Ct) ≤ C̃(C)Φ(t).

(7.13)

Let us next verify the invariance of the last assumption. Consider two selfadjoint reference operators P^{♯}, P^{♭} for the same operator P, on two different torii which both contain \overline{B(0, R_{0})}. Then P^{♯} and P^{♭} coincide near \overline{B(0, R_{0})}. We make all the general assumptions above for both operators, except (7.12), and we shall see in the end that if P^{♯} satisfies (7.12), then so does P^{♭}. After applying a diffeomorphism which is equal to the identity near \overline{B(0, R_{0})}, we may assume that T^{♯} = T^{♭} =: T so that P^{♯} and P^{♭} live in the same Hilbert space H^{♯} = H^{♭} and have the same domain D^{♯} = D^{♭}. Since we cannot take the diffeomorphism with Jacobian 1 in general, the two Hilbert spaces will carry different though equivalent norms uniformly with respect to h.

Let z vary in a bounded subset Ω of C \ R. Let Q^{♯}, Q^{♭} be elliptic differential operators on T, self-adjoint for the respctive inner products, which coincide with P^{♯} and P^{♭} respectively in T \ \overline{B(0, R_{0})}. Let ψ_{0} ≺ ψ_{1} ≺ ψ_{2} belong to C^{∞}(T; [0, 1]) with ψ_{0} equal to 1 near the closure of B(0, R_{0}) and with P^{♯} = P^{♭} near supp ψ_{2}. As an approximation for (z − P^{♭})^{−1}, we try

E^{♭} = ψ_{2}(z − P^{♯})^{−1}ψ_{1} + (1 − ψ_{0})(z − Q^{♭})^{−1}(1 − ψ_{1}).  (7.14)

(z −P^{♭})E^{♭}(z) = 1−[Q^{♯}, ψ_{2}](z −P^{♯})^{−1}ψ_{1} +[Q^{♯}, ψ_{0}](z −Q^{♭})^{−1}(1−ψ_{1}). (7.15)

For the last two terms we use telescopic formulas: Let θ_{0} ≺ θ_{1} ≺ .. ≺ θ_{N} be in C^{∞}(T) with supp θ_{N} ∩ \overline{B(0, R_{0})} = ∅. Iterating the identity

(z − P^{♯})^{−1}θ_{j} = θ_{j+1}(z − Q^{♯})^{−1}θ_{j} + (z − P^{♯})^{−1}[Q^{♯}, θ_{j+1}](z − Q^{♯})^{−1}θ_{j},

and similarly with P^{♯} replaced by Q^{♯}, and where θ_{j} may be replaced by [Q^{♯}, θ_{j}], we get

(z − P^{♯})^{−1}θ_{0} =

∑_{j=1}^{N} θ_{j}(z − Q^{♯})^{−1}[Q^{♯}, θ_{j−1}](z − Q^{♯})^{−1}[Q^{♯}, θ_{j−2}]..[Q^{♯}, θ_{1}](z − Q^{♯})^{−1}θ_{0}

+(z − P^{♯})^{−1}[Q^{♯}, θ_{N}](z − Q^{♯})^{−1}[Q^{♯}, θ_{N−1}]..[Q^{♯}, θ_{1}](z − Q^{♯})^{−1}θ_{0}.

(7.16)

The adjoint formula is:

θ_{0}(z − P^{♯})^{−1} =

∑_{j=1}^{N} θ_{0}(z − Q^{♯})^{−1}[θ_{1}, Q^{♯}]..[θ_{j−2}, Q^{♯}](z − Q^{♯})^{−1}[θ_{j−1}, Q^{♯}](z − Q^{♯})^{−1}θ_{j}

+θ_{0}(z − Q^{♯})^{−1}[θ_{1}, Q^{♯}]..[θ_{N−1}, Q^{♯}](z − Q^{♯})^{−1}[θ_{N}, Q^{♯}](z − P^{♯})^{−1}.

(7.17)

For the second term to the right in (7.15), take θ_{0}, .., θ_{N} as above with θ_{0} = 1 near supp[Q^{♯}, ψ_{2}] and with supp θ_{N} ∩ supp ψ_{1} = ∅. Then (7.17) gives

\begin{matrix}[Q^{♯}, ψ_{2}](z − P^{♯})^{−1}ψ_{1} = \\ ±[Q^{♯}, ψ_{2}]θ_{0}(z − Q^{♯})^{−1}[Q^{♯}, θ_{1}]..[Q^{♯}, θ_{N−1}](z − Q^{♯})^{−1}[Q^{♯}, θ_{N}](z − P^{♯})^{−1}ψ_{1}. \end{matrix}

(7.18)

Here ∥[Q^{♯}, θ_{N}](z−P^{♯})^{−1}ψ_{1}∥_{L(H,L^{2})} ≤ C/|Im z|, where we used the assumption (7.3) which implies the corresponding fact for P^{♯}. On the other hand,

\begin{matrix}[Q^{♯}, ψ_{2}]θ_{0}(z − Q^{♯})^{−1}[Q^{♯}, θ_{1}]...[Q^{♯}, θ_{N−1}](z − Q^{♯})^{−1} \\ = O_{N}(1)\frac{h^{N}}{|Im z|^{N}} : L^{2} → H^{N}, \end{matrix}

where we equip H^{N} with the h-dependent norm ∥⟨hD⟩^{N}u∥_{L^{2}}. Choosing N > n it follows that the last operator is of trace class as an operator in L^{2} with trace class norm bounded by O_{N}(1)h^{N−n}/|Im z|^{N}. We have then showed that [Q^{♯}, ψ_{2}](z − P^{♯})^{−1}ψ_{1} is trace negligible (cf.[80]) in the sense that for every N ∈ N there exists M(N) ≥ 0, such that

∥[Q^{♯}, ψ_{2}](z − P^{♯})^{−1}ψ_{1}∥_{tr} = O_{N}(1)\frac{h^{N}}{|Im z|^{M(N)}} .

Here ∥ · ∥_{tr} = ∥ · ∥_{C_{1}} denotes the trace class norm. Similarly we get the same fact for [Q^{♯}, ψ_{0}](z − Q^{♭})^{−1}(1 − ψ_{1}).

From (7.14), (7.15), we get

(z − P^{♭})^{−1} = ψ_{2}(z − P^{♯})^{−1}ψ_{1} + (1 − ψ_{0})(z − Q^{♭})^{−1}(1 − ψ_{1}) + R^{♭}(z), (7.19)

where R^{♭} is trace negligible. Actually we can replace (z−P^{♯})^{−1} by (z−P^{♭})^{−1}, changing only the trace negligible part. Similarly,

(z − P^{♯})^{−1} = ψ_{2}(z − P^{♯})^{−1}ψ_{1} + (1 − ψ_{0})(z − Q^{♯})^{−1}(1 − ψ_{1}) + R^{♯}(z), (7.20)

where R^{♯}(z) is trace negligible. If f ∈ C_{0}^{∞}(R), we have the simple Cauchy formula ([22]), much exploited since the work of [39]:

f(P^{♯}) = −\frac{1}{π} ∫ \frac{∂f̃}{∂\overline{z}}(z − P^{♯})^{−1}L(dz),  (7.21)

valid more generally for P^{♯} replaced by any self-adjoint operator. Here f̃ ∈ C_{0}^{∞}(C) is an almost analytic extension of f, i.e. with ∂f̃/∂\overline{z} vanishing to infinite order on R, and L(dz) is the Lebesgue measure on C. It follows from this and (7.19), (7.20), that

f(P^{♯}) − f(P^{♭}) = (1 − ψ_{0})(f(Q^{♯}) − f(Q^{♭}))(1 − ψ_{1}) + K,

where ∥K∥_{tr} = O(h^{∞}). On the other hand, we know that ∥f(Q^{♯})∥_{tr} , ∥f(Q^{♭})∥_{tr} are O(h^{−n}), so we conclude that

∥f(P^{♯}) − f(P^{♭})∥_{tr} = O(h^{−n}).  (7.22)

Let I_{1} ⊂⊂ I_{2} ⊂⊂ I_{3} ⊂⊂ R be fixed open intervals. Choosing f ∈ C_{0}^{∞}(R) first with 1_{I_{1}} ≺ f ≺ 1_{I_{2}} and then with 1_{I_{2}} ≺ f ≺ 1_{I_{3}}, we get

N(I_{1}, P^{♭}) − O(h^{−n}) ≤ N(I_{2}, P^{♯}) ≤ N(I_{3}, P^{♭}) + O(h^{−n}).  (7.23)

It is now clear that if P^{♯} satisfies (7.12) with λ = 2, then P^{♭} does so with λ = 1.

More generally, assume that P^{♯} satisfies (7.12) for all λ ≥ 1. Then we can view λ^{−1}P^{♯} as a reference operator associated to λ^{−1}P, and all assumptions above, including (7.12) are satisfied uniformly with h replaced by h̃ = h/\sqrt{λ}. The previous argument gives (7.12) for λ^{−1}P^{♭} with h replaced by h̃ and with λ replaced by 1/2. Making the inverse substitutions, we recover (7.12) for P^{♭}. #

7.2 Analytic distorsions.

This classical approach to resonances was initiated by Aguilar-Combes [1], Balslev-Combes [5], and then followed by many others, [70, 43]. Here we recall the approach of [80] with some minor modifications of [76], and we refer to these papers for more details and references.

A smooth (C^{∞}) manifold Γ ⊂ C^{n} is called totally real if

T_{x}Γ ∩ i(T_{x}Γ) = 0, ∀x ∈ Γ.  (7.24)

If Γ is totally real, then dim Γ ≤ n, and a natural example of a maximally totally real (m.t.r.) manifold (i.e. of maximal dimension n) is Γ = R^{n}. (The opposite extreme of totally real manifolds is given by complex manifolds, for which T_{x}Γ = iT_{x}Γ.) Totally real manifolds are mapped to totally real manifolds under holomorphic diffeomorphisms, so the notion extends to submanifolds of complex manifolds.

Let Γ ⊂ C^{n} be smooth of real dimension n. Locally we have Γ = f(R^{n}), where f : R^{n} → C^{n} is smooth with injective differential. Let f̃: C^{n} → C^{n} be an almost analytic extension of f so that \overline{∂}f̃ vanishes to infinite order on R^{n}. Let y ∈ R^{n}. Then since df̃ is complex linear, iT_{f(y)}Γ = (df̃(y))(iT_{y}R^{n}). Hence Γ is totally real near f(y) iff

df̃(y)(T_{y}R^{n}) ∩ df̃(y)(iT_{y}R^{n}) = 0,

i.e. iff df̃ is injective, or more explicitly iff

det (\frac{∂f_{j}(y)}{∂y_{k}}) ≠ 0  (7.25)

If Γ is m.t.r. and u ∈ C^{∞}(Γ), then locally we can find an almost analytic extension ũ ∈ C^{∞}(C^{n}) with ũ_{|}_{Γ} = u, \overline{∂}ũ = 0 to infinite order on Γ. If û is a second a.a. extension of the same function u, then ũ − û vanishes to ∞ order on Γ. In fact, let f be as above and let f̃ be an a.a. extension, so that f̃ is a local diffeomorphism according to (7.25). Then the inverse of f is a.a. on Γ and we can relate a.a. extensions from R^{n} and from Γ (locally) by means of f̃ and its inverse.

Let Ω ⊂ C^{n} be open, Γ ⊂ Ω a m.t.r. manifold. Let

P (x, D_{x}) = ∑_{|α|≤m} a_{α}(x)D_{x}^{α}  (7.26)

be an differential operator with holomorphic coefficient on Ω. We can define P_{Γ} : C^{∞}(Γ) → C^{∞}(Γ) by

P_{Γ}u = (P (x, D_{x})ũ)_{|}_{Γ},  (7.27)

where ũ is a local a.a. extension of u. If f is as above then

(P_{Γ}u) ∘ f = Q(y, D_{y})(u ∘ f),  (7.28)
Q(y, D_{y}) = ∑_{|α|≤m}(a_{α} ∘ f)(y)((^{t}∂_{y}f̃)^{−1}D_{y})^{α}.  (7.29)

In particular P_{Γ} is a differential operator of order m with smooth coefficients. The principal symbol q of Q is related to the principal symbol p of P by the usual relation

q(y, η) = p(f(y), (^{t}∂f̃(y))^{−1}η).

More invariantly, if we identify T^{∗}Γ with a submanifold of C^{n} × C^{n} via the map

T^{∗}Γ ∋ (x, dϕ(x)) ↦ (x, ∂_{x}ϕ̃(x)) ∈ Γ^{n} × C^{n},

for ϕ ∈ C^{∞}(C; R), then the principal symbol p_{Γ} of P_{Γ} is given by

p_{Γ} = p_{|}_{T^{∗}Γ}.  (7.30)

The quickest way to see this is to consider

p_{Γ}(x, dϕ(x)) = lim_{h→0} e^{−iϕ(x)/h}P_{Γ}(e^{iϕ(x)/h}).

This remark also carries over to principal symbols in the sense of h-differential operators: Let

P (x, hD_{x}) = ∑_{|α|≤m} a_{α}(x; h)(hD_{x})^{α}

with a_{α} holomorphic in Ω, a_{α} = a_{α}^{0}(x) + O(h), and define the corresponding principal symbol by

p(x, ξ) = ∑_{|α|≤m} a_{α}^{0}(x)ξ^{α}.

Correspondingly we can consider h-differential operators on manifolds and define their principal symbols in the sense of h-differential operators. (7.30) still holds.

We need a deformation result.

Lemma 7.2 Let ω ⊂ R^{n} be open and let f : [0, 1]×ω ∋ (t, y) ↦ f(t, y) ∈ C^{n} be a smooth proper map such that det(\frac{∂f}{∂y}) ≠ 0, ∀ (t, y) and such that f(t, ·) is injective. Assume that f(t, y) = f(0, y) for all y ∈ ω \K, where K is some compact subset of ω. Let P (x, D_{x}) be a differential operator with holomorphic coefficients defined in a neighborhood of f([0, 1] × ω) such that P_{Γ_{t}} is elliptic for 0 ≤ t ≤ 1, where Γ_{t} = f({t} × ω). If u_{0} ∈ D′(Γ_{0}) and P_{Γ_{0}}u_{0} extends to a holomorphic function in a neighborhood of f([0, 1] × ω), then u_{0} extends to a possibly multivalued holomorphic function in a neighborhood of the same set. More precisely, for every t ∈ [0, 1], there is a holomorphic function u_{t} defined in a neighborhood of Γ_{t} such that u_{t} = u_{s} near Γ_{s} when |t−s| is small enough, and u_{0} is an extension of u.

This result is standard and was proved in [80]. We follow that work:

Proof. We shall use an FBI type argument. Let Γ be a m.t.r. manifold. We use Lebeau’s resolution of the identity [51]: When Γ = R^{n}, we can represent the Dirac measure δ as

δ(x) = ∫ e^{i(x·ξ+iλ|ξ|x^{2}/2)}b_{λ}(x, ξ)dξ,  (7.31)

where λ > 0 and b_{λ}(x, ξ) = \frac{1}{(2π)^{n}} (1 + \frac{iλx·ξ}{2|ξ|}). This integral as well as the similar ones below should be interpreted as oscillatory integrals, i.e. as the limit in D′, when ϵ → 0 of

δ_{ϵ}(x) = ∫ e^{i(x·ξ+iλ|ξ|x^{2}/2)}b_{λ}(x, ξ)e^{−ϵξ^{2}/2}dξ,

and where the precise expression of the convergence factor exp(−ϵξ^{2}/2) is not of great importance, since it suffices to have a family χ_{ϵ}(ξ) of functions in S which is bounded in C_{b}^{∞} and converges to 1 uniformly on every compact set when ϵ → 0. This means that if u ∈ C_{0}^{∞}(R^{n}):

u(x) = ∫∫ e^{i((x−y)·ξ+iλ|ξ|(x−y)^{2}/2)}b_{λ}(x − y, ξ)u(y)dydξ.  (7.32)

Put ϕ_{α}(x, y, α) = (x − y) · α_{ξ} + iλ|α_{ξ}|((x − α_{x})^{2} + (y − α_{x})^{2}), a_{λ}(x, y, α) = c(λ)|α_{ξ}|^{\frac{n}{2}} b(x−y, α_{ξ}) for a suitable c(λ) > 0, α = (α_{x}, α_{ξ}). Then by evaluating a Gaussian integral in α_{x}, we get from (7.32) :

u(x) = ∫∫_{{y∈R^{n} \atop α∈R^{2n}}} e^{iϕ_{λ}(x,y,α)}a_{λ}(x, y, α)u(y)dydα.  (7.33)

We now pass to the case of general Γ, replacing |α_{ξ}| by its holomorphic extension \sqrt{α_{ξ}^{2}}, and work near some fixed point x_{0} ∈ Γ. We may assume x_{0} = 0, T_{x_{0}}^{∗}Γ = R^{n}. With λ > 0 sufficiently large, ϕ = ϕ_{λ}, a = a_{λ}, we put

Au(x) = ∫∫_{Γ×T^{∗}Γ} e^{iϕ(x,y,α)}a(x, y, α)χ_{1}(α_{x})χ(y)u(y)dydα, x ∈ Γ, (7.34)

where χ ∈ C_{0}^{∞}(Γ) is equal to 1 near 0 and χ ≺ χ_{1} ∈ C_{0}^{∞}(Γ). Notice that

Im ϕ ∼ |α_{ξ}|(|x − α_{x}|^{2} + |y − α_{x}|^{2}).

By Stokes’ formula, we can change the integration contour to y ∈ Γ, α_{x} ∈ Γ, α_{ξ} ∈ T_{y}^{∗}(Γ). (This is a change in α_{ξ} only, and we can use the intermediate contours given by y, α_{x} ∈ Γ, α_{ξ} ∈ T_{[ty+(1−t)α_{x}]}^{∗}Γ, where [z] denotes the point in Γ which is closest to z, for z in a small complex neighborhood of x_{0}.) Then we can integrate out α_{x} and get

Au(x) ≡ ∫_{y∈Γ} ∫_{ξ∈T_{y}^{∗}Γ} e^{i((x−y)·ξ+iλ|ξ|(x−y)^{2}/2)}b_{λ}(x − y, ξ)χ(y)u(y)dydξ =: Ãu(x),

(7.35)

modulo a term Ku(x), which extends holomorphically to a u-independent neighborhood of x_{0}. By analytic continuation from (7.32) we see that the distribution kernel of Ã vanishes outside the diagonal and since Ã is a pseudodifferential operator of order 0, we have

Ãu = ã(x)u(x).  (7.36)

Testing Ã on oscillatory functions and using Stokes’ formula, we see that ã = χ(x), so we end up with

χu(x) ≡ Au(x).  (7.37)

Using the ellipticity of P, we can construct an analytic symbol c(x, y, α), with α_{ξ} as the large parameter (see for instance [73]) of order \frac{n}{2} − m, defined in a conic neighborhood of {x_{0}, x_{0}} × T_{x_{0}}^{∗}Γ, holomorphic in x, y, α_{x} and with \overline{∂}_{α_{ξ}}c exponentially small (O(e^{−|α_{ξ}|/C}) for some C > 0), such that

tP (y, D_{y})(e^{iϕ(x,y,α)}c(x, y, α)) = e^{iϕ(x,y,α)}(a(x, y, α) − r(x, y, α)), (7.38)

where r is holomorphic in x, y and of exponential decrease:

|r(x, y, α)| ≤ Ce^{−|α_{ξ}|/C}.  (7.39)

Using this, we get for x in a neighborhood of x_{0} and assuming supp χ small:

Au(x) = ∫∫ e^{iϕ(x,y,α)}r(x, y, α)χ_{1}(α_{x})(χu)(y)dydα  (7.40)
+ ∫∫ e^{iϕ(x,y,α)}c(x, y, α)χ(y)χ_{1}(α_{x})P u(y)dydα
+ ∫∫ e^{iϕ(x,y,α)}c(x, y, α)χ_{1}(α_{x})[P, χ]u(y)dydα.

Notice that (^{t}P)_{Γ} = ^{t}(P_{Γ}).

The first and the last integrals extend holomorphically to a u-independent neighborhood of x_{0}. For the middle integral, we get the same conclusion, if we assume that P u extends to some fixed neighborhood Ω of x_{0}. To see this we notice that the exponential factor e^{iϕ(x,y,α)} become exponentially small, if we make a small α-dependent deformation of the integration in y near y = x_{0} and restrict x to sa small complex neighborhood of that point.

Summing up, we have proved: Let P be holomorphic near Γ with P_{Γ} elliptic. If Γ is m.t.r., x_{0} ∈ Γ, u ∈ D′(Γ) and P u extends to some neighborhood Ω of x_{0}, then u extends to another neighborhood Ω̃ of x_{0}, which does not depend on u. Also Ω̃ will not collapse, if we vary Γ reasonably. From this the lemma follows #

We now return to the operators in section 7.1 and add an assumption:

There exist θ_{0} ∈]0, π[, ϵ > 0, R_{1} ≥ R_{0}, such that the  (7.41)
coefficients a_{α}(x; h) of Q extend holomorphically in x to

{rω; ω ∈ C^{n}, dist (ω, S^{n−1}) < ϵ, r ∈ e^{i[0,θ_{0}]}]R_{1}, +∞[}, and the relevant parts of (7.6), (7.8) remain valid

in the sense that a_{α} and all its x-derivatives are bounded for x in this larger set, uniformly in h, and the convergence in (7.8) remains valid when x tends to infinity in the larger set above, again uniformly in h.

For given ϵ_{0} > 0, R_{1} > R_{0}, we can construct a smooth function [0, π] × [0, ∞[∋ (θ, t) ↦ f_{θ}(t) ∈ C, injective for every θ, with the properties

(i) f_{θ}(t) = t for 0 ≤ t ≤ R_{1},

(ii) 0 ≤ arg f_{θ}(t) ≤ θ, ∂_{t}f_{θ} ≠ 0,

(iii) arg f_{θ}(t) ≤ arg ∂_{t}f_{θ}(t) ≤ arg f_{θ}(t) + ϵ_{0},

(iv) f_{θ}(t) = e^{iθ}t, for t ≥ T_{0}, where T_{0} only depends on ϵ_{0} and R_{1}.

Consider the map

κ_{θ} : R^{n} ∋ x = tω ↦ f_{θ}(t)ω ∈ C^{n}, t = |x|.

The image Γ_{θ} is a m.t.r. manifold which coincides with R^{n} along B(0, R_{1}). We choose R_{1} at least as large as R_{1} in the assumption (7.41).

Let H_{θ} = H_{R_{0}} ⊕ L^{2}(Γ_{θ} \ B(0, R_{0})). By means of κ_{θ} we can identify H_{θ} with H and similarly we can define D_{θ} = D. Let χ ∈ C_{0}^{∞}(B(0, R_{1})) be equal to 1 near \overline{B(0, R_{0})} and define the unbounded operator P_{θ} : H → H with domain D by

P_{θ}u = P (χu) + Q_{Γ_{θ}}(1 − χ)u.

Here we assume that 0 ≤ θ ≤ θ_{0}.

Parametrizing Γ_{θ} by means of κ_{θ} we get:

−Δ_{Γ_{θ}} = (\frac{1}{f′(t)}D_{t})^{2} − \frac{n − 1}{f(t)f′(t)}iD_{t} + (f(t))^{−2}D_{ω}^{2}, f = f_{θ},

where −D_{ω}^{2} denotes the Laplacian on S^{n−1}. If ω^{∗}^{2} denotes the principal symbol of D_{ω}^{2} and τ is the dual variable of t, then the principal symbol of −Δ_{Γ_{θ}} is

p_{0,θ} = \frac{τ^{2}}{f′(t)^{2}} + \frac{ω^{∗2}}{f(t)^{2}} ,  (7.42)

so pointwise −Δ_{Γ_{θ}} is elliptic (if ϵ_{0} > 0 is small enough) and the principal symbol p_{0,θ} takes its values in a sector of angle ≤ 2ϵ_{0} and globally it takes its values in the sector

−2(θ + ϵ_{0}) ≤ arg p_{0,θ} ≤ 0.  (7.43)

Since θ ≤ θ_{0}, this sector is of angle < 2π (when ϵ_{0} is small enough).

Choosing R_{1} large enough, we get

In R^{n} \ B(0, R_{0}), h^{−2}P_{θ} is an elliptic differential operator (7.44)

whose principal symbol in the classical sense over each fixed point in Γ_{θ} takes its values in a sector of angle ≤ 3ϵ_{0}, and globally in a sector − 2θ − 3ϵ_{0} ≤ arg z ≤ ϵ_{0}

The coefficients of P_{θ} − e^{−2iθ}(−h^{2}Δ_{R^{n}})  (7.45)

(as a semiclassical differential operator) tend to 0 uniformly w.r.t. h, when Γ_{θ} ∋ x → ∞, and we identify Γ_{θ} and R^{n} by means of κ_{θ}.

Lemma 7.3 If z ∈ C\{0}, arg z ≠ −2θ, then P_{θ} −z : D → H is a Fredholm operator of index 0.

The proof [75] is a minor modification of the corresponding one in [80]:

Proof. This has nothing to do with the smallness of h, so we assume h = 1 for simplicity. We first show that P_{θ} − z is a Fredholm operator, and for that it suffices to invert P_{θ} − z modulo compact operators. On Γ_{θ} ≃ R^{n}, we introduce a smooth partition of unity :

1 = θ_{1} + θ_{2} + θ_{3},  (7.46)

where 1_{B(0,R_{0})} ≺ θ_{1} ∈ C_{0}^{∞}(B(0, R_{1})), θ_{2} ∈ C_{0}^{∞}, supp θ_{3} ⊂ {x ∈ R^{n}; |x| ≥ R}, R ≫ 1. Further, 0 ≤ θ_{j} ≤ 1. Let χ_{1}, χ_{2}, χ_{3} have the same properties as θ_{j} with the exception of (7.46) and with θ_{j} ≺ χ_{j}. Let z_{0} ∈ C \ R, and put

E(z) = χ_{1}(P − z_{0})^{−1}θ_{1} + QQ̃θ_{2} + χ_{3}(−e^{−2iθ}Δ − z)^{−1}θ_{3},

where Q̃ is a properly supported parametrix of P_{θ} − z (i.e. a pseudodifferential operator which inverts the latter operator modulo smoothing operators outside a small neighborhood of the closure of B(0, R_{0}) with the property that the distribution kernel is supported in a thin neighborhood of the diagonal). Here θ_{3}, χ_{3} have their support in the region where Γ_{θ} = e^{iθ}R^{n}. Then we get

(P_{θ} − z)E(z) = 1 + K(z),

with

K(z) = (z_{0} − z)χ_{j}(P − z_{0})^{−1}θ_{1} + [P_{θ}, χ_{1}](P − z_{0})^{−1}θ_{1} + ((P_{θ} − z)Q̃ − 1)θ_{2}
+[P_{θ}, χ_{3}](−e^{−2iθ}Δ − z)^{−1}θ_{3} + χ_{3}(P_{θ} − (−e^{−2iθ}Δ))(−e^{−2iθ}Δ − z)^{−1}θ_{3}
= K′(z) + K′′(z),

where K′′(z) is the last term in the middle member. With R large enough, we can arrange so that ∥K′′(z)∥_{L(L^{2},L^{2})} < 1/2. On the other hand, K′(z) is compact. Now write

(P_{θ} − z)E(z)(1 + K′′(z))^{−1} = 1 + K′(1 + K′′)^{−1},

to get a right inverse modulo a compact operator.

As an approximate left inverse, we put

F (z) = θ_{1}(P − z_{0})^{−1}χ_{1} + θ_{2}Q̃ + θ_{3}(−e^{−2iθ}Δ − z)^{−1}χ_{3}.

Then

F (P_{θ} − z) = 1 + L(z), L(z) = L′(z) + L′′(z),  (7.47)

where

L′(z) = I + II + III + IV ,
L′′ = θ_{3}(−e^{−2iθ}Δ − z)^{−1}(P_{θ} − (−e^{−2iθ}Δ))χ_{3},
I = (z_{0} − z)θ_{1}(P − z_{0})^{−1}χ_{1},
II = −θ_{1}(P − z_{0})^{−1}[P, χ_{1}],
III = θ_{2}(Q̃(P_{θ} − z) − 1),
IV = −θ_{3}(−e^{−2iθ}Δ − z)^{−1}[P_{θ}, χ_{3}].

We can arrange so that ∥L′′∥_{L(D_{θ},D_{θ})} is a small as we like, and we turn to the proof that L′ is compact: D_{θ} → D_{θ}.

Clearly, III, IV have this compactness property. For I and II, we first see that they are bounded: D_{θ} → D_{θ}, so it is enough to show that (P − z_{0})I, (P − z_{0})II are compact: D_{θ} → H_{θ}. We have

(P − z_{0})I = (z − z_{0})θ_{1} + (z_{0} − z)[P, θ_{1}](P − z_{0})^{−1}χ_{1},

where θ_{1} is compact: D → H, and [P, θ_{1}](P − z_{0})^{−1}χ_{1} is even compact as an operator : H → H. Similarly,

(P − z_{0})II = \underbrace{θ_{1}[P, χ_{1}]}_{=0} − \underbrace{[P, θ_{1}](P − z_{0})^{−1}[P, χ_{1}]}_{compact:D→H}

is compact. Hence L′ is compact D_{θ} → D_{θ} and we invert (P_{θ} − z) to the left modulo a compact operator D_{θ} → D_{θ}, by means of (1 + L′′)^{−1}F :

(1 + L′′)^{−1}F (P_{θ} − z) = 1 + (1 + L′′)^{−1}L′.

We have then showed that P_{θ} − z : D_{θ} → H_{θ} is Fredholm. This operator depends contiuously on (θ, z), so the index is constant under deformation in (θ, z) with z ∉ e^{−2iθ}[0, ∞[. Deforming to θ = 0, z = i, we see that the index is 0. #

Corollary 7.4 A point z in C\e^{−2iθ}[0, ∞[ belongs to σ(P_{θ}) iff N (P_{θ}−z) ≠ 0.

Proposition 7.5 Assume that 0 ≤ θ_{1} < θ_{2} ≤ θ_{0}, and let z_{0} ∈ C\e^{−2i[θ_{1},θ_{2}]}[0, ∞[. Then

dim N (P_{θ_{2}} − z_{0}) = dim N (P_{θ_{1}} − z_{0}).

Proof. We follow [80] ([75]). Again the statement does not depend on the smallness of h, so we take h = 1 for simplicity. We shall prove that the two kernels are ”equal” via holomorphic extension. We shall apply Lemma 7.2 to suitable deformations of the contours. Let χ ∈ C_{0}^{∞}(]\frac{1}{2}, \frac{3}{2}[; [0, 1]) be equal to 1 near 1 and put for T ≥ 1:

F_{θ_{1},θ_{1},T} (t) = f_{θ_{1}}(t) + χ(\frac{t}{T})(f_{θ_{2}}(t) − f_{θ_{1}}(t)).

Let Γ_{θ_{1},θ_{2},T} be the corresponding submanifold of C^{n}. If θ_{2} − θ_{1} is small enough, Lemma 7.2 is applicable to the family of m.t.r. manifolds,

[0, 1] ∋ t ↦ Γ_{θ_{1},θ_{1}+t(θ_{2}−θ_{1}),T} .

We conclude that if (P_{θ_{1}} − z_{0})u_{θ_{1}} = 0, u ∈ D_{θ_{1}}, then u_{θ_{1}} extends to a holomorphic function in a neighborhood of ∪_{θ_{1}≤θ≤θ_{2}}(Γ_{θ} \ B(0, R_{1})), and by restriction, we get an element u_{θ} ∈ D_{θ}^{loc} with (P_{θ} − z_{0})u_{θ} = 0.

We also need to control u_{θ} near infinity, and for that we shall use the assumption on z_{0} which assures ellipticity near ∞.

Let 0 < a_{1} < a_{2} < b_{2} < b_{1}, Ω_{j} = {z ∈ R^{n}; a_{j} < |z| < b_{j}}, j = 1, 2. The change of variables y = T z transforms −Δ_{y} into −\frac{1}{T^{2}} Δ_{z}. We take T ≥ 1 and view the latter operator as semi-classical one with h = 1/T, and use the standard a priori estimate

∑_{|α|≤2} ∥(T^{−1}D_{z})^{α}u∥_{L^{2}(Ω_{2})} ≤ C(∥(−T^{−2}e^{−2iθ}Δ_{z} − z_{0})u∥_{L^{2}(ω_{1})} + ∥u∥_{L^{2}(Ω_{1}\Ω_{2})}),

with C independent of T. In the y-variables this becomes

∑_{|α|≤2} ∥D_{y}^{α}u∥_{L^{2}(T Ω_{2})} ≤ C(∥(−e^{−2iθ}Δ_{y} − z_{0})u∥_{L^{2}(T Ω_{1})} + ∥u∥_{L^{2}(T (Ω_{1}\Ω_{2}))}). (7.48)

For θ_{2} −θ_{1} small enough and T large enough, and for t/T near supp χ, we may view P_{Γ_{θ_{1},θ_{2},T}} as a small perturbation of −e^{−2iθ_{1}}Δ_{y}. We conclude that

∥u_{θ_{2}}∥_{H^{2}(T Ω)} ≤ C∥u_{θ_{1}}∥_{L^{2}(T V)}.

Now cover a neighborhood of ∞ in Γ_{θ_{2}} by a sequence T_{k}Ω, where T_{k} = 2^{k}T_{0}. If u_{θ_{1}} ∈ D_{θ_{1}}, it follows that u_{θ_{2}} ∈ D_{θ_{2}}.

To sum up, if θ_{2} − θ_{1} is small enough, we have constructed an injective linear map (by holomorphic extension and restriction): N (P_{θ_{1}} − z_{0}) → N (P_{θ_{2}} − z_{0}) The inverse map is constructed the same way, so dim N (P_{θ_{1}} − z_{0}) = dim N (P_{θ_{2}} − z_{0}).

When θ_{2} − θ_{1} is larger, we introduce intermediate θ-values and apply the above. #

Two extensions of the lemma are possible:

1) We may consider inhomogeneous equations for (P_{θ} − z) assuming suitable holomorphic extension properties of the right hand side.

2) We may consider N ((P_{θ} − z)^{k}) for k ∈ N.

The preceding result shows that for 0 ≤ θ_{1} ≤ θ ≤ θ_{2} ≤ θ_{0}, the spectrum of P_{θ} in C \ e^{−2i[θ_{1},θ_{2}]}[0, +∞[ is independent of θ. In particular P_{θ} will have no spectrum in a small sector e^{i]0,ϵ[}]0, +∞[. Analytic Fredholm theory shows that the spectrum of P_{θ} in C \ e^{−2iθ}[0, +∞[ is discrete. In particular for θ = 0, we see that the spectrum in ] − ∞, 0[ is discrete. If 0 < θ < \frac{π}{2}, the spectrum of P_{θ} outside e^{−2iθ}[0, +∞[ consists of the negative eigenvalues of P_{0} and the eigenvalues in e^{−i[0,2θ[}]0, +∞[. When θ ≥ \frac{π}{2}, we only get the latter eigenvalues. By definition, the resonances of P are the eigenvalues of P_{θ} in the sector e^{−2i[0,θ[}]0, +∞[, and as we have seen, they are independent of θ in the sense that replacing θ by a larger value will not change the set of resonances in the sector e^{−2i[0,θ[}]0, +∞[.

By analytic Fredholm theory, if z_{0} ∈ e^{−i[0,2θ[}]0, +∞[, 0 ≤ θ ≤ θ_{0} is a resonance, then the spectral projection

π_{θ,z_{0}} = \frac{1}{2πi} ∫_{γ(z_{0})}(z − P_{θ})^{−1}dz  (7.49)

is of finite rank, where γ(z_{0}) denotes the positively oriented boundary of a small disc centered at z_{0}, which contains no other resonances than z_{0}. This rank is independent of the choice of θ for which z_{0} belongs to the sector where the resonances are defined, and by definition it is the multiplicity of the resonance z_{0}. The image F_{θ,z_{0}} is contained in the domain of any power of P and is invariant under P_{θ}. (P_{θ} − z_{0})_{|}_{F_{θ,z_{0}}} is nilpotent and F_{θ,z_{0}} = N ((P_{θ}−z_{0})^{k_{0}}) for some k_{0} ∈ N. The proof of Proposition 2.4 gives a bijection F_{θ_{1},z_{0}} → F_{θ_{2},z_{0}} under the assumptions there.

To end this chapter we mention that resonances defined by means of complex distorsions, coincide with other definitions. (See [37] for general results in this direction.) In particular, one can show that the resolvent (z − P)^{−1} : H_{comp} → D_{loc}, defined first for Im z > 0, can be extended meromorphically across ]0, +∞[ to the sector e^{−i[0,2θ_{0}[}]0, +∞[ and the poles are just the resonances. [[[Add the proof of this statement]]]. The multiplicity of a resonance is the rank of the formal spectral projection, obtained by replacing the resolvent of P_{θ} in the integral (7.49) by the meromorphic extension of the resolvent of P. See [80].

The set of resonances will be denoted by Res P and the elements of this set will be counted (repeated) according to their multiplicity.

8 The local trace formula and local upper bounds

8.1 Some trace estimates

Let P_{·} : H → H, · = 0, 1, be two selfadjoint operators satisfying the assumptions in the preceding section. Here

H_{·} = H_{·,R_{0}} ⊕ L^{2}(R^{n} \ B(0, R_{0}))  (8.1)

is an orthogonal sum. In particular for the reference operators P_{·}^{♯}, we have corresponding estimates #(σ(P_{·}) ∩ [−λ, λ]) ≤ O(1)Φ_{·}(\frac{λ}{h^{2}} ). In view of (7.13), we notice that Φ_{·} are of at most polynomial growth:

Φ_{·}(t) = O(t^{n_{·}/2})  (8.2)

for some exponents n_{·} ∈ [n, +∞[.

To the earlier assumptions, we add:

|[a_{·,α}(x; h)]_{0}^{1}| ≤ O(1)⟨x⟩^{−_{n}}, for some ñ > n  (8.3)

not only in the real domain but also in the complex domain appearing in (7.41). Here θ_{0} > 0 is assumed to be the same for both operators P_{1}, P_{0}, and we write [a_{·}]_{0}^{1} = a_{1} − a_{0}.

Proposition 8.1 Let f ∈ C_{0}^{∞}(R) be independent of h or vary in a bounded subset of C_{0}^{∞}(R). Let χ ∈ C_{0}^{∞}(R^{n}) be equal to 1 near \overline{B(0, R_{0})}. Then χf(P_{·}), f(P_{·})χ, [(1 − χ)f(P_{·})(1 − χ)]_{0}^{1} are of trace class and

′′tr [f(P_{·})]_{0}^{1}′′ :=  (8.4)

[tr (χf(P_{·})χ + χf(P_{·})(1 − χ) + (1 − χ)f(P_{·})χ)]_{0}^{1} + tr [(1 − χ)f(P_{·})(1 − χ)]_{0}^{1}

is independent of the choice of χ and is O(1)Φ_{max} (h^{−2}). Here Φ_{max} := max(Φ_{0}, Φ_{1}).

Let ϵ_{0} > 0 be fixed and small enough, so that 2θ_{0} +ϵ_{0} < 2π. Let W ⊂⊂ Ω be open relatively compact subsets of e^{i]−2θ_{0},ϵ_{0}]}]0, +∞[. We assume that these sets are independent of h and that Ω is simply connected. Also assume that the intersections J_{+}, I_{+} between Ω, W and ]0, +∞[ are intervals. Let Ω_{−}, W_{−} denote the intersections of e^{i]−2θ_{0},0]}]0, +∞[ with Ω and W respectively. The local trace formula ([75, 76]) is given in the following result:

Theorem 8.2 Fix Ω, W as above. Then for h > 0 sufficiently small the following holds: Let f = f(z; h) be holomorphic for z ∈ Ω and satisfy |f(z; h)| ≤ 1, for z ∈ Ω \ W. Let χ ∈ C_{0}^{∞}(J_{+}) be independent of h, with χ = 1 near \overline{I}_{+}. Then uniformly with respect to f:

′′tr [(χf)(P_{·})]_{0}^{1}′′ =
[  ∑_{λ∈Res P_{·}∩W_{−}} f(λ)]_{0}^{1} − [  ∑_{μ∈σ(P_{·})∩]−∞,0[∩W_{−}} f(μ)]_{0}^{1} + O(1)Φ_{max}(\frac{1}{h^{2}} ).

(8.5)

The proof also shows that the number of resonances of P_{·} in Ω_{−} is O(1)Φ_{·}(h^{−2}).

Later in these notes we will also discuss the more classical trace formula of Bardos-Guillot-Ralston [6] with an important improvement of R. Melrose [55] and further generalized in [85]. That formula relates all the resoances to the trace of the wave-group for operators as in Chapter 2, when the dimension is odd. In the cited works, it is proved by using the Lax–Phillips theory and in particular translation represenations, but following the work [35], Zworski also gave a proof in [102] which uses more general scattering theory and which is closer to the proof that we shall give for the local trace formula (following [76]).

It should also be pointed out that the left hand side in the local trace formula is the same as in the Birman–Krein formula for the scattering phase. See [...].

In this section we shall prove Proposition 8.1, by adapting arguments from [80] and the local trace formula will be proved in the next section. When working with the P_{·} separately, we generally drop the subscript “·”. If χ is as in the Proposition, choose a torus T for the definition of a reference operator P^{♯} as in Chapter 7, and choose χ̃ ∈ C_{0}^{∞}(R^{n}) with χ ≺ χ̃. We may arrange so that supp χ̃ is naturally included in T and P^{♯} = P near suppχ̃. Recall that trace negligible operators were introduced in Chapter 7.

Lemma 8.3 The following operators are trace negligible for z in any fixed bounded set in C \ R:

\begin{matrix}χ(z − P)^{−1}χ − χ(z − P^{♯})^{−1}χ, \\ (1 − χ)(z − P)^{−1}(1 − χ) − (1 − χ)(z − Q)^{−1}(1 − χ), \\ χ(z − P)^{−1}(1 − χ̃), \\ χ(z − Q)^{−1}(1 − χ̃). \end{matrix}  (8.6)

Proof. We start with χ(z − P)^{−1}(1 − χ̃). Let χ ≺ θ_{N} ≺ ... ≺ θ_{1} ≺ χ̃, θ ∈ C_{0}^{∞}. Then we have the telescopic formula:

(z − P)^{−1}(1 − χ̃) =

∑_{j=1}^{N}(1 − θ_{j})(z − Q)^{−1}[Q, 1 − θ_{j−1}]...(z − Q)^{−1}[Q, 1 − θ_{1}](z − Q)^{−1}(1 − χ̃)

+(z − P)^{−1}[Q, 1 − θ_{N}](z − Q)^{−1}[Q, 1 − θ_{N−1}]...[Q, 1 − θ_{1}](z − Q)^{−1}(1 − χ̃),

so

\begin{matrix}χ(z − P)^{−1}(1 − χ̃) = \\ χ(z − P)^{−1}[Q, 1 − θ_{N}](z − Q)^{−1}[Q, 1 − θ_{N−1}]...[Q, 1 − θ_{1}](z − Q)^{−1}(1 − χ̃). \end{matrix}

We conclude as in the proof of the invariance of (7.12), and see that χ(z − P)^{−1}(1−χ̃) is trace negligible. The operator χ(z−Q)^{−1}(1−χ̃) can be treated in the same way.

Let χ_{0} ∈ C_{0}^{∞}(R^{n}) be equal to one near \overline{B(0, R_{0})} and satisfy χ_{0} ≺ χ. Write

(z−P)^{−1}(1−χ) = (1−χ_{0})(z−Q)^{−1}(1−χ)−(z−P)^{−1}[Q, χ_{0}](z−Q)^{−1}(1−χ),

so that

\begin{matrix}(1 − χ)(z − P)^{−1}(1 − χ) − (1 − χ)(z − Q)^{−1}(1 − χ) = \\ −((1 − χ)(z − P)^{−1}χ_{1})([Q, χ_{0}](z − Q)^{−1}(1 − χ)), \end{matrix}

where we inserted a cutoff χ_{1} ∈ C_{0}^{∞} with χ_{0} ≺ χ_{1} ≺ χ, and indicated a factorization of the last term into 2 factors. The first of the two factors is trace negligible, while the second one is bounded of norm O(|Im z|^{−1}), so the operator above is trace negligible.

The argument for χ(z − P)^{−1}χ − χ(z − P^{♯})^{−1}χ is the same except that we use χ̃ instead of 1 − χ_{0} and χ instead of 1 − χ. #

Let χ_{1} ∈ C_{0}^{∞}(R^{n}) with χ ≺ χ_{1}, and choose the reference torus large enough, to include supp χ_{1}. Consider the short telescopic formula:

(z − P)^{−1}χ = χ_{1}(z − P^{♯})^{−1}χ + (z − P)^{−1}[P, χ_{1}](z − P^{♯})^{−1}χ.

The proof of Lemma 8.3 shows that [P, χ_{1}](z − P^{♯})^{−1}χ is trace negligible, so the same holds for (z − P)^{−1}[P, χ_{1}](z − P^{♯})^{−1}χ.

Let f ∈ C_{0}^{∞}(R). Then,

f(P)χ = χ_{1}f(P^{♯})χ + O(h^{∞})

in trace norm. The trace norm of χ_{1}f(P^{♯})χ is O(1)Φ(h^{−2}), so the same holds for f(P)χ and by duality for χf(P).

From the lemma it follows that the trace class norm of (1−χ)f(P)(1−χ)− (1 − χ)f(Q)(1 − χ) is O(h^{∞}), so in order to prove that the trace class norm of [(1 − χ)f(P_{·})(1 − χ)]_{0}^{1} is O(h^{−n}), as we shall, it suffices to prove the same fact about [f(Q_{·})]_{0}^{1}.

Let 1 = ∑_{j∈Z^{n}} χ_{j}(x) be a partition of unity with χ_{j}(x) = χ_{0}(x−j) ∈ C_{0}^{∞}. For N ∈ N, consider

(i + Q_{1})^{−N}(z − Q_{1})^{−1} − (i + Q_{0})^{−N}(z − Q_{0})^{−1},  (8.7)

which is a linear combination of terms

(i + Q_{1})^{−k}(Q_{1} − Q_{0})(i + Q_{0})^{−(N+1−k)}(z − Q_{0})^{−1}  (8.8)

for 1 ≤ k ≤ N, and of the term

(i + Q_{1})^{−N}(z − Q_{1})^{−1}(Q_{1} − Q_{0})(z − Q_{0})^{−1}.  (8.9)

For (8.8), we consider

χ_{ν}(i + Q_{1})^{−k}(Q_{1} − Q_{0})χ_{ρ}(i + Q_{0})^{−(N+1−k)}(z − Q_{0})^{−1}χ_{μ}.  (8.10)

Using a Combes-Thomas argument, (see [66], [39], [75], section 7) [explain..] we see that this operator is bounded: L^{2} → H^{2(N+1)} of norm

O(1)\frac{1}{|Im z|}⟨ρ⟩^{−_{n}}e^{− \frac{1}{Ch} ((|ν−ρ|−C)_{+}+|Im z|(|ρ−μ|−C)_{+})},

with t_{+} = max(t, 0), t ∈ R.

(8.11)

If N ≥ N(n) is large enough, this is also a bound for h^{n} times the trace norm for operators L^{2} → L^{2}, and summing over ν, μ, we get

\begin{matrix}∥(i + Q_{1})^{−k}(Q_{1} − Q_{0})χ_{ρ}(i + Q_{0})^{−(N+1−k)}(z − Q_{0})^{−1}∥_{tr} \\ = O(h^{−n})⟨ρ⟩^{−_{n}} \frac{1}{|Im z|}(1 + \frac{h}{|Im z|})^{n}. \end{matrix}  (8.12)

For (8.9) we get similarly

\begin{matrix}∥(i + Q_{1})^{−N}(z − Q_{1})^{−1}(Q_{1} − Q_{0})χ_{ρ}(z − Q_{0})^{−1}∥_{tr} \\ = O(h^{−n})⟨ρ⟩^{−_{n}} \frac{1}{|Im z|^{2}} (1 + \frac{h}{|Im z|})^{2n}. \end{matrix}  (8.13)

Summing over ρ, we get

\begin{matrix}∥(i + Q_{1})^{−N}(z − Q_{1})^{−1} − (i + Q_{0})^{−N}(z − Q_{0})^{−1}∥_{tr} \\ = O(h^{−n})\frac{1}{|Im z|^{2}} (1 + \frac{h}{|Im z|})^{2n}. \end{matrix}  (8.14)

For f ∈ C_{0}^{∞}(R), we write f(t) = (t + i)^{−N}g(t) with g ∈ C_{0}^{∞}(R). Then

\begin{matrix}[f(Q_{·})]_{0}^{1} = [(i + Q_{·})^{−N}g(Q_{·})]_{0}^{1} \\ = −\frac{1}{π} ∫ \frac{∂_{g}}{∂\overline{z}}(z)[(i + Q_{·})^{−N}(z − Q_{·})^{−1}]_{0}^{1}L(dz), \end{matrix}  (8.15)

where g̃ ∈ C_{0}^{∞}(C) is an almost analytic extension of g. Using (8.14), we get

∥[f(Q_{·})]_{0}^{1}∥_{tr} = O(h^{−n}),  (8.16)

and we conclude (as already mentioned) that

∥[(1 − χ)f(P_{·})(1 − χ)]_{0}^{1}∥_{tr} = O(h^{−n}).  (8.17)

We have then proved everything in Proposition 8.1, except that the right hand side of (8.4) is independent of the choice of χ, which we leave to the reader. #

Remark. Let f ∈ C_{0}^{∞}(]−∞, 0[). Possibly after increasing R_{0}, we may assume that (z − Q)^{−1} exists and is equal to O(1) for z in some fixed complex neighborhood of supp f so that f(Q) = 0. Lemma 8.3 together with the operator Cauchy formula then shows that (1 − χ)f(P)(1 − χ) is trace negligible, now in the sense that the trace class norm is O(h^{∞}). Choose χ_{0} ∈ C_{0}^{∞}(R^{n}) with 1_{B(0,R_{0})} < χ_{0} < χ. Then (1 − χ_{0})f(P)(1 − χ_{0}) is also trace negligible and

tr f(P) = tr χf(P)χ + tr (1 − χ)f(P)χ + tr χf(P)(1 − χ) + O(h^{∞}) .

Here tr χf(P)(1−χ) = tr χf(P)(1−χ)(1−χ_{0})^{2} = tr χ(1−χ_{0})f(P)(1−χ_{0})(1− χ) = O(h^{∞}). Similarly tr (1 − χ)f(P)χ = O(h^{∞}) so tr f(P) = tr χf(P)χ + O(h^{∞}) Similarly tr f(P^{♯}) = tr χf(P^{♯})χ + O(h^{∞}) and from Lemma ltrace1.3 and the Cauchy formula we have trχf(P)χ = tr χf(P^{♯})χ + O(h^{∞}). We conclude that

tr f(P) = tr f(P^{♯}) + O(h^{∞}) .

Remark. If P_{0}, P_{1} satisfy the assumptions of Proposition ltrace1.1, then uniformly with respect to λ ≥ 1, the same holds for the operators λ^{−1}P_{0}, λ^{−1}P_{1}, provided that we replace h by the new semi-classical parameter h/\sqrt{λ}. The same holds for Theorem 8.2. In particular, if P_{0}, P_{1} are independent of h and satisfy all assumptions of Proposition ltrace1.1 (or Theorem ltrace1.2) which make sense for h = 1, then λ^{−1}P_{0}, λ^{−1}P_{1} do the same with the semiclassical parameter h = 1/\sqrt{λ}.

8.2 Proof of the local trace formula.

We choose m ∈ N large enough to get nice trace class properties for some operators below. We take θ = θ_{0}. Let Ω_{+} be the intersection of Ω with the sector 0 ≤ arg z < ϵ_{0} (for some ε_{0} > 0), define W_{+} similarly and recall that W_{−}, Ω_{−} were defined prior to Theorem 8.2. Fix a point z_{0} ∈ e^{i[3ϵ_{0},min (π,2π−2θ−3ϵ_{0})]}]0, +∞[ away from σ(P_{·}) and σ(P_{·,θ}) (where we use the observation that we can always replace ϵ_{0} by some smaller number without changing anything in the conclusion of Theorem 8.2) and write

f(z) = (z − z_{0})^{−m}g(z).  (8.18)

Then

(χf)(P_{·}) = (P_{·} − z_{0})^{−m}(χg)(P_{·}) =  (8.19)

−\frac{1}{π} ∫ g(z)\frac{∂χ̃}{∂\overline{z}}(z)(P_{·} − z_{0})^{−m}(z − P_{·})^{−1}L(dz),

where χ̃ ∈ C_{0}^{∞}(Ω) is an almost analytic extension of χ with support in a small neighborhood of J_{+} and equal to 1 near \overline{I}_{+}. We first look at

I_{·}^{−} := −\frac{1}{π} ∫_{Ω_{−}} g(z)\frac{∂χ̃}{∂\overline{z}}(z)(P_{·} − z_{0})^{−m}(z − P_{·})^{−1}L(dz).  (8.20)

Let χ̂ ∈ C_{0}^{∞}(Ω) be equal to 1 near \overline{W}_{−}, equal to χ̃ near J_{+} and be almost analytic near R_{−}, in case Ω intersects that set. If Ω∩R_{−} = ∅, we can replace χ̃ by χ̂ in the last formula. More generally, we have

I_{·}^{−} = −(χ̂1_{R_{−}}f)(P_{·}) − \frac{1}{π} ∫_{Ω_{−}} g(z)\frac{∂χ̂(z)}{∂\overline{z}}(P_{·} − z_{0})^{−m}(z − P_{·})^{−1}L(dz). (8.21)

The support of ∂χ̂(z)/∂\overline{z} in Ω_{−} does not intersect W_{−}, so |f| ≤ 1 there and consequently g = O(1) uniformly w.r.t. h on that set. Choosing m large enough, it follows from the first remark after (8.17) and the proof of Proposition 8.1, that

′′tr [I_{·}^{−}]_{0}^{1}′′ = −[  ∑_{μ∈σ(P_{·})∩R_{−}∩W} f(μ)]_{0}^{1} + O(1)Φ_{max}(\frac{1}{h^{2}} ).  (8.22)

Next look at

I_{·}^{+} := −\frac{1}{π} ∫_{Ω_{+}} g(z)\frac{∂χ̂(z)}{∂\overline{z}}(P_{·} − z_{0})^{−m}(z − P_{·})^{−1}L(dz).

Green’s formula gives for every δ > 0:

I_{·}^{+} = −\frac{1}{π} ∫_{Ω_{+}∩\{Im z≤δ\}} g(z)\frac{∂χ̂(z)}{∂\overline{z}}(P_{·} − z_{0})^{−m}(z − P_{·})^{−1}L(dz) (8.23)

+\frac{1}{2πi} ∫_{Ω_{+}∩\{Im z=δ\}} g(z)χ̂(z)(P_{·} − z_{0})^{−m}(z − P_{·})^{−1}dz,

where the integration contour in the last integral is oriented in the direction of decreasing Re z. If δ > 0 is small enough (independent of h), the integrand in the first integral in (8.23) has its support in the region where g = O(1) and as in the proof of Proposition 8.1, we get

′′tr [I_{·}^{+}]_{0}^{1}′′ = ′′tr [J_{·}]_{0}^{1}′′ + O(1)Φ_{max}(\frac{1}{h^{2}} ),  (8.24)

where J_{·} denotes the last integral in (8.23).

Committing another error O(1)Φ_{max}(\frac{1}{h^{2}} ) in the evaluation of ′′tr [J_{·}]_{0}^{1}′′, we may replace J_{·} by

J_{·} := \frac{1}{2πi} ∫_{γ} g(z)(P_{·} − z_{0})^{−m}(z − P_{·})^{−1}dz,  (8.25)

where γ is the segment from b to a, where Im b = Im a = δ, Re a, Re b ∈ J_{+}, Re a < inf I_{+}, Re b > sup I_{+}.

Proposition 8.4 We have

′′tr [J_{·}]_{0}^{1}′′ = ′′tr [J̃_{·}]_{0}^{1}′′,  (8.26)

where

J̃_{·} = \frac{1}{2πi} ∫_{γ} g(z)(P_{·,θ} − z_{0})^{−m}(z − P_{·,θ})^{−1}L(dz),  (8.27)

and where P_{·,θ} = P_{·}_{|Γ_{θ}} is defined as in section 3, with the parameter ϵ_{0} there chosen small enough depending on δ.

Proof. Take χ ∈ C^{∞}(R; [0, 1]) with χ(t) = 0 for t ≤ 0, χ(t) = 1 for t ≥ 1. As in the proof of Proposition 7.5 (but now with a different χ), we put

f_{θ_{1},θ_{2},T} (t) = f_{θ_{1}}(t) + χ(\frac{t}{T})(f_{θ_{2}}(t) − f_{θ_{1}}(t)),

and let Γ_{θ_{1},θ_{2},T} be the corresponding contour. As in Chapter 7 we assume that 0 ≤ θ_{1} < θ_{2} ≤ θ with θ_{2} − θ_{1} small enough, and we let P_{·,θ_{1},θ_{2},T} denote the restriction of P_{·} to Γ_{θ_{1},θ_{2},T}. The proof of Proposition 8.1 can be applied to show that

′′tr [(P_{·,θ_{1},θ_{2},T} − z_{0})^{−m}(z − P_{·,θ_{1},θ_{2},T})^{−1}]_{0}^{1}′′  (8.28)

is O(1)Φ_{max}(\frac{1}{h^{2}} ), depends continuously on T > 0, is equal to

′′tr [(P_{·,θ_{2}} − z_{0})^{−m}(z − P_{·,θ_{2}})^{−1}]_{0}^{1}′′

for small T, and tends to

′′tr [(P_{·,θ_{1}} − z_{0})^{−m}(z − P_{·,θ_{1}})^{−1}]_{0}^{1}′′,

when T → +∞.

To get the proposition, it suffices to show that the expression (8.28) is independent of T. Let Γ_{t} = {f(t, y); y ∈ ω}, 0 ≤ t ≤ 1, be a smooth family of m.t.r. manifolds in C^{n}, where ω is open and f in C^{∞}, with det(\frac{∂f}{∂y}) ≠ 0 everywhere.

If u is holomorphic near Γ_{t} for some t, let u_{t}(y) = u(f(t, y)), so that u_{t} is the restriction u_{|}_{Γ_{t}}, expressed in the parametrization f(t, ·). Then

\frac{∂u_{t}(y)}{∂t} = (\frac{∂f}{∂t} (t, y) · \frac{∂u}{∂x})(f(t, y)).  (8.29)

Since Γ_{t} is m.t.r., there exists a complex vector field ν_{t}(y, \frac{∂}{∂y}) on Γ_{t} (tangent to Γ_{t}), such that

\frac{∂u_{t}}{∂t} = ν_{t}u_{t}.  (8.30)

Moreover ν_{t} depends smoothly on (t, y) and vanishes outside the support of \frac{∂f}{∂t} (t, ·).

Let P be a differential operator with holomorphic coefficients, defined near Γ_{t}. Define P_{t} = P_{Γ_{t}} as in section 3, so that

P_{t}u_{t} = (P u)_{t},  (8.31)

when u is holomorphic near Γ_{t}. Then

ν_{t}(P_{t}u_{t}) = ∂_{t}(P_{t}u_{t}) = (∂_{t}P_{t})(u_{t}) + P_{t}∂_{t}u_{t} = (∂_{t}P_{t})(u_{t}) + P_{t}ν_{t}u_{t},

and since the functions of the type u_{t} are “sufficiently dense”, we conclude that in the sense of differential operators,

∂_{t}P_{t} = [ν_{t}, P_{t}].  (8.32)

This applies to Γ_{θ_{1},θ_{2},T} and passes to the resolvents and their compositions:

\begin{matrix}∂_{T} ((P_{·,θ_{1},θ_{2},T} − z_{0})^{−m}(z − P_{·,θ_{1},θ_{2},T})^{−1}) = \\ [ν, (P_{·,θ_{1},θ_{2},T} − z_{0})^{−m}(z − P_{·,θ_{1},θ_{2},T})^{−1}], \end{matrix}  (8.33)

where ν = ν_{θ_{1},θ_{2},T} is a smooth complex vector field whose support is compact and disjoint from \overline{B(0, R_{0})}.

When considering (8.28), we may choose the cutoff χ in (8.4) with support disjoint from that of ν, so with ψ equal to χ or to 1 − χ,

tr χ[ν, (P_{·,θ_{1},θ_{2},T} − z_{0})^{−m}(z − P_{·,θ_{1},θ_{2},T})^{−1}]ψ =
tr χν ∘ (...)^{−m}(..)^{−1}ψ − tr χ(...)^{−m}(..)^{−1} ∘ νψ
= 0 − tr (...)^{−m}(..)^{−1}ν ∘ ψχ = 0,

where we used the cyclicity of the trace. The T -derivative of the expression (8.28) then reduces to the following expression, where we drop θ_{1}, θ_{2}, T for simplicity:

\begin{matrix}tr ([(1 − χ)[ν, (P_{·} − z_{0})^{−m}(z − P_{·})^{−1}](1 − χ)]_{0}^{1}) \\ = tr [ν, [(1 − χ)(P_{·} − z_{0})^{−m}(z − P_{·})^{−1}(1 − χ)]_{0}^{1}]. \end{matrix}  (8.34)

Here [(1−χ)(P_{·} −z_{0})^{−m}(z −P_{·})^{−1}(1−χ)]_{0}^{1} is of trace class also as an operator L^{2} → H^{1}, so the expression (8.34) vanishes by the cyclicity of the trace. It follows that the T -derivative of (8.28) is zero and the proof is complete. #

It remains to study “tr [J̃_{·}]_{0}^{1}”. We shall use a finite rank perturbation of P_{·,θ}. Let F be a smooth mapping from a neighborhood of e^{i[−2(θ+ϵ_{0}),ϵ_{0}]}[0, ∞[ into itself such that

F (z) = z, for |z| large and for z near e^{−2iθ}[0, ∞[,  (8.35)
\overline{Ω} is disjoint from the image of F.  (8.36)

If θ < \frac{π}{2}, we further extend F to be a smooth map from a neighborhood of e^{i[−π−ϵ_{0},ϵ_{0}]}[0, ∞[ to itself, by putting F (z) = z for z near e^{i[−π−ϵ_{0},−2θ]}[0, ∞[.

In the appendix (with a suitable choice of Γ_{θ}) we shall construct an operator P̂_{·,θ} : D_{·} → H_{·} with the following properties:

K_{·} := P̂_{·,θ} − P_{·,θ} is of rank O(1)Φ_{max}(\frac{1}{h^{2}} ) and
O(1) : D(P_{·}^{N}) → D(P_{·}^{M}) for all N, M ∈ N.

(8.37)

Notice here that D(P_{·}^{0}) = H_{·}.

K_{·} is compactly supported in the sense that K_{·} = χK_{·}χ (8.38)

if χ ∈ C_{0}^{∞} is equal to 1 on B(0, R) for some sufficiently large R.

For every N ∈ N, (P̂_{·,θ} − z)^{−1} = O(1) : D(P_{·}^{N}) → D(P_{·}^{N+1}), (8.39)
uniformly for z ∈ \overline{Ω}.

Since we shall always work with operators on Γ_{θ} in the remainder of the proof, we drop the subscript θ most of the time and write P_{·} instead of P_{·,θ}.

Write

z − P_{·} = (1 + K̃_{·})(z − P̂_{·}), where K̃_{·} = K_{·}(z − P̂_{·})^{−1},  (8.40)

so that

(z − P_{·})^{−1} = (z − P̂_{·})^{−1}(1 + K̃_{·})^{−1}.  (8.41)

Using the resolvent identity,

(z − P_{·})^{−1} − (z − P̂_{·})^{−1} = −(z − P_{·})^{−1}K_{·}(z − P̂_{·})^{−1},

we can decompose the right hand side of (8.26) as I + II, where

I = ′′tr [\frac{1}{2πi} ∫_{γ} g(z)(P_{·} − z_{0})^{−m}(z − P̂_{·})^{−1}dz]_{0}^{1}′′,  (8.42)

II = −′′tr [\frac{1}{2πi} ∫_{γ} g(z)(P_{·} − z_{0})^{−m}(z − P_{·})^{−1}K_{·}(z − P̂_{·})^{−1}dz]_{0}^{1}′′. (8.43)

Let γ̃ be a smooth contour from b to a (as was the case with γ) and contained in Ω \ W. The integrand in I is holomorphic in Ω, so we may replace γ by γ̃ in (8.42). Along γ̃, we have g = O(1) (uniformly w.r.t. h) and if we choose a cutoff χ in Lemma 8.3 equal to 1 on a sufficiently large bounded (sub)set (of Γ_{θ}), we can apply the proof of Proposition 8.1, to obtain:

′′tr [(P_{·} − z_{0})^{−m}(z − P̂_{·})^{−1}]_{0}^{1}′′ = O(1)Φ_{max}(\frac{1}{h^{2}} ).  (8.44)

We therefore get

I = O(1)Φ_{max}(\frac{1}{h^{2}} ).  (8.45)

Since K_{·} are of finite rank, we can write

II = −[tr \frac{1}{2πi} ∫_{γ} g(z)(P_{·} − z_{0})^{−m}(z − P_{·})^{−1}K_{·}(z − P̂)^{−1}dz]_{0}^{1}.  (8.46)

Here we can replace (P_{·} − z_{0})^{−m} by (z − z_{0})^{−m} because the difference has an integrand which is holomorphic in Ω, and the contour γ can then be replaced by γ̃. It follows that the difference gives a contribution O(1)Φ_{max}(\frac{1}{h^{2}} ) to II. Recalling (8.18), we get:

II = −[\frac{1}{2πi} ∫_{γ} f(z)tr ((z − P_{·})^{−1}K_{·}(z − P̂_{·})^{−1})dz]_{0}^{1} + O(1)Φ_{max}(\frac{1}{h^{2}} ). (8.47)

Now use (8.41) and the cyclicity of the trace:

−tr ((z − P_{·})^{−1}K_{·}(z − P̂_{·})^{−1}) = −tr ((z − P̂_{·})^{−1}(1 + K̃_{·})^{−1}K_{·}(z − P̂_{·})^{−1})

= −tr ((1 + K̃_{·})^{−1}K_{·}(z − P̂_{·})^{−2}) = tr ((1 + K̃_{·}(z))^{−1} \frac{∂}{∂z} K̃_{·}(z)).

(8.48)

Remark. In the last equality in (8.48) we used that K_{·} is independent of z. It may be of interest to see what would happen if K_{·} = K_{·}(z) (and P̂_{·} = P̂_{·}(z)) depends holomorphically on z. Differentiating the first identity in (8.40), we get

−K̃_{·}(z − P̂_{·})^{−1} = \frac{∂K̃_{·}}{∂z} − (1 + K̃_{·})\frac{∂P̂_{·}}{∂z} (z − P̂_{·})^{−1}.  (8.49)

Using this in the third expression in (8.48), we get

−tr ((z − P_{·})^{−1}K_{·}(z − P̂_{·})^{−1}) =  (8.50)

tr ((1 + K̃_{·})^{−1} \frac{∂K̃_{·}}{∂z}) − tr ((1 + K̃_{·})^{−1}(1 + K̃_{·})\frac{∂P̂_{·}}{∂z} (z − P̂_{·})^{−1})

= tr ((1 + K̃_{·})^{−1} \frac{∂K̃_{·}}{∂z}) − tr(\frac{∂P̂_{·}}{∂z} (z − P̂_{·})^{−1}).

Here the last term is holomorphic in Ω and we get (8.51) below also in this more general case.

Substitution of (8.48) into (8.47) gives

II = [\frac{1}{2πi} ∫_{γ} f(z)tr ((1 + K̃_{·}(z))^{−1} \frac{∂K̃_{·}}{∂z})dz]_{0}^{1} + O(1)Φ_{max}(\frac{1}{h^{2}} ).  (8.51)

With (8.40) in mind we consider an identity of the form

A(z) = B(z)C(z),  (8.52)

where C(z) : H_{1} → H_{2}, B(z) : H_{2} → H_{3} depend holomorphically on z in some domain, and ∂_{z}B(z) is of trace class. Restricting the attention to some subdomain, where B and C have bounded inverses, we get after differentiating the equation (8.52) and using the cyclicity of the trace:

tr (B^{−1}∂_{z}B) = tr (A^{−1}∂_{z}A − C^{−1}∂_{z}C).  (8.53)

Let Γ be a relatively compact subdomain with smooth boundary and assume that C^{−1} exists everywhere on the closure of Γ. Also assume that B(z)^{−1} exists everywhere on the boundary of Γ. Then for f holomorphic in Γ and continuous up the boundary, we get

tr (\frac{1}{2πi} ∫_{∂Γ} f(z)B(z)^{−1}∂B(z)dz) =  (8.54)

tr \frac{1}{2πi} ∫_{∂Γ} f(z)(A^{−1}∂_{z}A − C^{−1}∂_{z}C)dz = tr (\frac{1}{2πi} ∫_{∂Γ} f(z)A^{−1}∂_{z}Adz).

We apply this to (8.40) with Γ ⊂⊂ Ω and ∂Γ avoiding the resonances, and get:

tr \frac{1}{2πi} ∫_{∂Γ} f(z)(1 + K̃_{·}(z))^{−1}∂_{z}K̃_{·}dz = ∑_{λ∈Res (P_{·})∩Γ} f(λ).  (8.55)

Notice that we could have taken the trace inside the integral, that tr ((1 + K̃_{·}))^{−1}∂_{z}K̃_{·}) = ∂_{z} log det(1+K̃_{·}(z)) and that the resonances are precisely the zeros of det(1 + K̃_{·}(z)) and that the multiplicities agree.

The remainder of the proof will have common features with a proof of Lidskii’s theorem, close to the one in [31], of which we gave a variant in Chapter 5. Put

D_{·}(z; h) := det(1 + K̃_{·}) = O(1)e^{O(1)Φ_{·}(h^{−2})}.  (8.56)

Since we will work separately with the two operators P_{·}, we drop the subscript · most of the time in the following. In Ω_{+,δ} := {z ∈ Ω_{+}; Im z > δ} with δ > 0 (suitably chosen and independent of h), not only (z−P̂) : D → H is uniformly invertible, but also (z − P), and hence

(1 + K̃)^{−1} = (z − P̂)(z − P)^{−1} = O(1) : H → H.

It follows that (1 + K̃)^{−1} = 1 + (P − P̂)(z − P)^{−1} with

∥(P − P̂)(z − P)^{−1}∥_{tr} = O(1)Φ(\frac{1}{h^{2}} ),

giving an upper bound for the determinant of (1 + K̃)^{−1} as in (8.56):

|D(z; h)| ≥ e^{−CΦ(h^{−2})}, z ∈ Ω_{+,δ}.  (8.57)

Let N = N(P, Ω; h) be the number of resonances in Ω counted with their multiplicity. (8.56) remains valid in a slightly larger domain and combining this with (8.57) and Jensen’s inequality, we get

N(P, Ω; h) ≤ CΦ(\frac{1}{h^{2}} ).  (8.58)

At this point, we could apply an estimate of H. Cartan about lower bounds of holomorphic functions outside small unions of circles around the zeros as in [81], but we prefer to keep the argument of [75, 76], which is simple, more or less standard and gives precisely what is needed.

Let z_{j}, j = 1, .., N be the resonances in Ω repeated according to their multiplicity and put

D_{w}(z; h) = ∏_{j=1}^{N}(z − z_{j}).  (8.59)

Thanks to (8.58), we have

|D_{w}(z; h)| ≤ e^{CΦ(h^{−2})} in Ω, |D_{w}(z; h)| ≥ e^{−CΦ(h^{−2})} in Ω_{+,δ}.  (8.60)

In Ω \ Ω_{+,δ} we can get the same lower bound, if we avoid to go too close to the z_{j}. For that, we first establish the simple

Lemma 8.5 Let x_{1}, .., x_{N} ∈ R and let I ⊂ R be an interval of length |I| ∈ ]0, +∞[. Then there exists x ∈ I, such that

∏_{j=1}^{N} |x − x_{j}| ≥ e^{−N(1+log \frac{2}{|I|})}.

Proof. Consider F (x) = ∑_{j=1}^{N} log \frac{1}{|x−x_{j}|}. We have

∫_{I} log \frac{1}{|x − x_{j}|}dx ≤ 2 ∫_{0}^{|I|/2} log \frac{1}{t}dt = |I|(1 + log \frac{2}{|I|}),

since the first integral takes its largest possible value when x_{j} is the midpoint of I. It follows that

∫_{I} F (x)dx ≤ N|I|(1 + log \frac{2}{|I|}).

We can therefore find x ∈ I, such that F (x) ≤ N log(1 + log \frac{2}{|I|}), i.e.

∏_{j=1}^{N} |x − x_{j}| = e^{−F (x)} ≥ e^{−N(1+log \frac{2}{|I|})}.

#

Let γ̃ = γ̃_{α}, α ∈ J be a smooth family of smooth curves joining b to a, where J is a bounded interval of length ≠ 0. We let γ̃_{α} move transversally in Ω_{+,δ/2} when α varies in such a way that for every z ∈ Ω_{−} we have:

1) If z belongs to γ̃_{α} for some α, then this α is unique, α = α(z) and dist (z, γ̃_{β}) ≥ |β − α(h)| for all β ∈ J.

2) If z belongs to no γ̃_{β}, then dist (z, γ̃_{β}) ≥ dist (β, ∂J) for every β ∈ J. It follows from Lemma 8.5, that if I ⊂ J is an interval of length > 0, then

we can find α ∈ I such that

|D_{w}(z; h)| ≥ e^{−CΦ(h^{−2})}, z ∈ γ̃_{α},  (8.61)

where C = C_{|I|}. We factorize D:

D(z; h) = G(z; h)D_{w}(z; h) z ∈ Ω,  (8.62)

where G and 1/G are holomorphic in Ω. Combining (8.60), (8.61), (8.62), we get

|G(z; h)| ≤ e^{CΦ(h^{−2})}, z ∈ Ω_{+,δ} ∪ γ̃_{α}.  (8.63)

The maximum principle gives

|G(z; h)| ≤ e^{CΦ(h^{−2})}, z ∈ Ω̃,  (8.64)

where Ω̃ ⊂⊂ Ω is any simply connected relatively open h-independent set with \overline{W} in its (relative) interior. In fact, it suffices to choose the family γ̃_{α} with γ̃_{α} ∩ Ω̃ ∩ Ω_{−} = ∅.

The relations (8.60, (8.57) imply that

|G(z; h)| ≥ e^{−CΦ(h^{−2})}, z ∈ Ω_{+,δ}.  (8.65)

Choose C > 0 large enough, so that

ℓ(z; h) := CΦ(\frac{1}{h^{2}} ) − log |G(z; h)| ≥ 0.  (8.66)

Notice that ℓ is harmonic. Harnack’s inequality tells us that for every K ⊂⊂ Ω̃, there is a constant C_{K} ≥ 1 such that for every non-negative harmonic function ℓ̃ on Ω̃ :

sup_{K} ℓ̃≤ C_{K} inf_{K} ℓ̃,  (8.67)

This applies to ℓ and after an arbitrarily small decrease of Ω̃, we have (8.67) with K = Ω̃, ℓ̃= ℓ and if we use (8.65), we get ℓ(z; h) ≤ CΦ(\frac{1}{h^{2}} ) on Ω_{+,δ} and hence by (8.67) (with K = Ω̃):

ℓ(z; h) ≤ CΦ(\frac{1}{h^{2}} ) on Ω̃.  (8.68)

We conclude that log |G| ≥ −CΦ(\frac{1}{h^{2}} ) on Ω̃ and with (8.64) we get

| log |G(z; h)|| ≤ CΦ(\frac{1}{h^{2}} ), z ∈ Ω̃.  (8.69)

Since log |G(z; h)| = Re log G is harmonic, we get after an arbitrarily small decrease of Ω̃:

∇(Re log G) = O(1)Φ(\frac{1}{h^{2}} ), z ∈ Ω̃.  (8.70)

The Cauchy Riemann equations for log G imply

\frac{d}{dz} log G = O(1)Φ(\frac{1}{h^{2}} ), z ∈ Ω̃.  (8.71)

Let γ̃ be a curve from b to a in Ω̃ \ W and which avoids the resonances. From (8.58), (8.59) we get

∫_{γ} f(z; h)\frac{d}{dz} log D_{w}(z; h)dz = O(1)Φ(\frac{1}{h^{2}} ).  (8.72)

Since

\frac{d}{dz} log D = \frac{d}{dz} log G + \frac{d}{dz} log D_{w},

we get from (8.71), (8.72):

∫_{γ} f(z; h)\frac{d}{dz} log D dz = O(1)Φ(\frac{1}{h^{2}} ).  (8.73)

We finally return to (8.51), where we recall (8.55) and the subsequent identity. Together with (8.73), this gives:

II =  (8.74)

[ ∑_{Res(P·)∋z between_{γ}and γ} f(z; h) + \frac{1}{2πi} ∫_{γ} f(z; h)\frac{d}{dz} (log D_{·}(z))dz]_{0}^{1} + O(1)Φ_{max}(\frac{1}{h^{2}} )

= [ ∑_{Res(P·)∋z between_{γ}and γ} f(z; h)]_{0}^{1} + O(1)Φ_{max}(\frac{1}{h^{2}} ).

Here we reintroduce the subscripts “.” and recall that Φ_{max} = max(Φ_{0}, Φ_{1}).

Since the total number of z_{j}’s is O(1)Φ(\frac{1}{h^{2}} ) and |f| ≤ 1 for z ∈ Ω \ W, we may replace the last sum in (8.74) by

∑_{z∈W_{−}∩Res(P_{·})} f(z; h)

and the proof of Theorem 8.2 is complete. #

Remark. Applying Lemma 8.3 or rather the equivalent estimate on ∫_{I} F (x)dx in its proof, we see that if we take a family of contours γ̃_{α}, α ∈ J, |J| > 0, as earlier in the proof, then (for every h) we can find α ∈ J, such that (8.73) strengthens to

∫_{γ_{α}} |\frac{d}{dz} log D||dz| = O(1)Φ(\frac{1}{h^{2}} ).  (8.75)

In fact, in view of (8.71) it suffices to establish this with D replaced by D_{w}.

Appendix: construction of P̂_{·,θ}.

We shall construct an operator P̂_{·,θ}, which satisfies (8.37)–(8.39) and for that we shall use the function F which was introduced prior to those equations. Since we shall work with the two operators P_{·} separately we drop the subscript ·. Let p_{θ} be the semi-classical principal symbol of P_{θ}, defined as ∑_{|α|≤2} b_{α}(x; h)ξ^{α} mod O(h⟨ξ⟩/⟨x⟩), where P_{θ} = ∑_{|α|≤2} b_{α}(x; h)(hD_{x})^{α}. Choosing Γ_{θ} suitably (more precisely with R_{1} sufficiently large and ϵ_{0} sufficiently small in the construction of f_{θ} in section 3), we see that F ∘ p_{θ} is a well-defined symbol with values away from \overline{Ω}, such that F ∘ p_{θ} = p_{θ}, when |x| ≥ R_{2}, R_{2} > R_{1}. Here we recall that Γ_{θ} coincides with R^{n} for |x| ≤ R_{1}.

Let P^{♯} be a reference operator as in section 2 and assume that the corresponding torus T is large enough so that B(0, R_{1}) is naturally contained in T. If f = F_{|}_{R}, then

f(P^{♯}) = P^{♯} + g(P^{♯}) = P^{♯} + K^{♯},  (A.1)

where f(E) = E + g(E), g ∈ C_{0}^{∞}(R), and

K^{♯} = O(1) : H^{♯} → H^{♯}, rank K^{♯} = O(1)Φ(\frac{1}{h^{2}} ).  (A.2)

More generally K^{♯} is uniformly O(1) as a bounded operator from the domain of any power of P^{♯} to the domain of any other power of P^{♯}. Moreover

(z − f(P^{♯}))^{−1} = O(1),  (A.3)

uniformly for z ∈ Ω and in h. Let Q^{♯} be a self-adjoint differential operator as in the construction of P^{♯}, so that Q^{♯} = P^{♯} in T \ B(0, R_{0}). As in the proof of Lemma 8.3, we see that if χ ∈ C_{0}^{∞}(T \ \overline{B(0, R_{0})}), then

χ(z − P^{♯})^{−1}χ − χ(z − Q^{♯})^{−1}χ is O(\frac{h^{N}}{|Im z|^{M(N)}} ) : H^{−N} → H^{N}, ∀N ∈ N.

(A.4)

Using the Cauchy formula (7.21) for g(P^{♯}), g(Q^{♯}), we see that

χf(P^{♯})χ − χf(Q^{♯})χ = O(h^{N}) : H^{−N} → H^{N}, ∀N ∈ N  (A.5)

for χ as in (A.4).

Using a result of Helffer-Robert, which can also be proved with the Cauchy formula (7.21) (see [21] and further references there), we know that χf(Q^{♯})χ is a h-pseudo-differential operator on T of the natural class, with leading symbol χ(x)^{2}f(p^{♯}(x, ξ)), where p^{♯} is the semiclassical leading symbol of Q^{♯}.

Let 1 = χ_{0} + χ_{1} + χ_{2}, where χ_{0} ∈ C_{0}^{∞}(B(0, R_{1}); [0, 1]) is equal to 1 near \overline{B(0, R_{0})}, χ_{1} ∈ C_{0}^{∞}(Γ_{θ}; [0, 1]), χ_{0} + χ_{1} = 1 near \overline{B(0, R_{2})}. Then χ_{2} ∈ C_{b}^{∞}(Γ_{θ}; [0, 1]) has its support disjoint from \overline{B(0, R_{2})}. Let χ_{j} ≺ χ̃_{j}, where χ̃_{j} ∈ C^{∞}(Γ_{θ}; [0, 1]) has its support close to that of χ_{j}, and define

P̃_{θ} := χ̃_{0}f(P^{♯})χ_{0} + χ̃_{1}R_{F} χ_{1} + χ̃_{2}P_{θ}χ_{2},  (A.6)

where R_{F} is an h-pseudodifferential operator with leading symbol F (p_{θ}) such that the total (Weyl) symbol (with Γ_{θ} identified with R^{n}) of R_{F} − P_{θ} (for x near supp χ_{1}) has compact support in ξ.

Lemma 8.6 If h > 0 is small enough, then z − P̃_{θ} is invertible for every z ∈ Ω and, for every N ∈ N, the inverse satisfies

(z − P̃_{θ})^{−1} = O_{N}(1) : D^{N} → D^{N+1},  (A.7)

uniformly for z ∈ Ω, where D^{N} is the domain of P^{N}, so that D^{0} = H.

Proof. The operator z − P̃_{θ} : D → H is Fredholm of index 0, for z ∈ Ω, so to get invertibility and the estimate (A.7) in the case N = 0, it is enough to show the a priori estimate:

∥u∥^{2} ≤ C∥(z − P̃_{θ})u∥^{2},  (A.8)

for u ∈ D. Let ψ_{0}, ψ_{1}, ψ_{2} ∈ C_{b}^{∞}(C_{θ}; [0, 1]) have the same support properties as χ_{0}, χ_{1}, χ_{2} and with

1 = ψ_{0}^{2} + ψ_{1}^{2} + ψ_{2}^{2}.  (A.9)

Assume that ψ_{0} ≺ χ_{0}. Then for some r_{1} > 0 and for all z ∈ Ω

∥(z − P̃_{θ})ψ_{0}u∥^{2} = ∥(z − f(P^{♯}))ψ_{0}u∥^{2} + O(h^{∞})∥u∥^{2},

and for j = 1, 2:

∥(z − P̃_{θ})ψ_{j}u∥^{2} ≥ r_{1}^{2}∥ψ_{j}u∥^{2}.

Then

∥(z − P̃_{θ})u∥^{2} = ∑_{0}^{2} ∥ψ_{j}(z − P̃_{θ})u∥^{2} ≥ ∑_{0}^{2}(∥(z − P̃_{θ})ψ_{j}u∥ − ∥[ψ_{j}, P̃_{θ}]u∥)^{2}

≥ r_{1}^{2}∥u∥^{2} − O(h)∥u∥_{D}^{2}.

But

∥u∥_{D_{θ}} ≤ O(1)(∥(z − P̃_{θ})u∥ + ∥u∥),

so we get (A.8) and the stronger a priori estimate

∥u∥_{D} ≤ C∥(z − P̃_{θ})u∥,  (A.10)

which is precisely (A.7) for N = 0.

If (z − P̃_{θ})u = v, with u, v ∈ D, then

(z_{0} − P_{θ})u = v + ((z_{0} − z) + (P̃_{θ} − P_{θ}))u ∈ D,  (A.11)

and applying (z − P̃_{θ}) and using (A.10), we see that ∥(z_{0} − P_{θ})u∥_{D} ≤ C∥v∥_{D}. This gives the case N = 1 of (A.7) and we can continue this argument by iteration. #

Write

P̃_{θ} = P_{θ} + χ̃_{0}g(P^{♯})χ_{0} + χ̃_{1}(R_{F} − P_{θ})χ_{1}.

We can find T_{F} of finite rank O(h^{−n}) and = O(1) : H^{s_{1}} → H^{s_{2}} for all s_{1}, s_{2} ∈ R such that

χ̃_{1}((R_{F} − P_{θ}) − T_{F})χ_{1} = O(h^{∞}) : H^{s_{1}} → H^{s_{2}}

for all s_{1}, s_{2} ∈ R. Put

P̂_{θ} = P_{θ} + χ̃_{0}g(P^{♯})χ_{0} + χ̃_{1}T_{F} χ_{1}.  (A.12)

Then it easy to see that (8.37)-(8.39) hold.

9 Clouds of resonances for −h^{2}Δ + V (x)

The applications that we develop in this chapter exploit very much the semiclassical nature of the operator. In chapter 10 we shall get other applications, also for operators that are not semi-classical to start with, and which exploit the existence of closed classical trajectories. The proof of the results in this chapter will also use a semi-classical trace formula of D. Robert, and we start by discussing certain distributions on the real axis that appear in connection with that formula.

Let V_{0}, V_{1} be continuous real-valued functions on R^{n} which tend to 0, when |x| → ∞, and satisfy

|V_{1}(x) − V_{0}(x)| ≤ C⟨x⟩^{−_{n}},  (9.1)

where ñ > n.

For E > 0, let ν_{+,·}(E) = ∫_{V_{·}(x)≥E} dx. This is a decreasing function of E, so μ_{+,·}(E) := −\frac{d}{dE} ν_{+,·}(E) is a positive measure on ]0, +∞[ with support equal to ]0, sup V_{·}]. Similarly, for E < 0, we put ν_{−,·}(E) = ∫_{V_{·}(x)≤E} dx (an increasing function of E,) μ_{−,·}(E) = \frac{d}{dE} ν_{−,·}(E) which is a positive measure on ] − ∞, 0[ with support in [inf V_{·}, 0[.

For ϕ ∈ C_{0}^{∞}(R), we put

⟨μ, ϕ⟩ = ∫ [ϕ ∘ V_{·}]_{0}^{1}dx = ∫ (ϕ(V_{1}(x)) − ϕ(V_{0}(x)))dx

Clearly μ is a distribution of order ≤ 1 with support contained in [min inf V_{·}, max sup V_{·}], ∫ μ(E)dE = 0, and

μ_{|}_{R_{±}} = [μ_{±,·}]_{0}^{1}.  (9.2)

For f ∈ C_{0}^{∞}(R), we put

⟨ω, f⟩ = ∫∫ [f(ξ^{2} + V_{·}(x))]_{0}^{1}dxdξ,

so that ω is a distribution on R of order ≤ 1 with

supp ω ⊂ [min inf V_{·}, +∞[.

We have with E_{+} := max(E, 0) and ∗ denoting convolution:

∫ f(ξ^{2} + V_{·}(x))dξ = ∫_{0}^{∞} f(E + V_{·}(x))d(Vol \{ξ ∈ R^{n}; |ξ| ≤ \sqrt{E}\})

= Vol (B_{R^{n}}(0, 1)) ∫ f(V_{·}(x) + E)d(E_{+}^{n/2})

= \frac{n}{2}Vol (B_{R^{n}}(0, 1)) ∫ f(V_{·}(x) + E)E_{+}^{\frac{n}{2} −1}dE

= C_{n}(f ∗ (−·)_{+}^{\frac{n}{2} −1})(V_{·}(x)),

where C_{n} > 0. Then

⟨ω, f⟩ = C_{n}⟨μ, f ∗ (−·)_{+}^{\frac{n}{2} −1}⟩ = C_{n}⟨μ ∗ (·)_{+}^{\frac{n}{2} −1}, f⟩,

so

ω = C_{n}μ ∗ (·)_{+}^{\frac{n}{2} −1}.  (9.3)

In general, if α > −1 and if H_{α}(x) = x_{+}^{α}, then Ĥ_{α}(ξ) = C_{α}(ξ − i0)^{−1−α}, where C_{α} ≠ 0. Let E_{α} ∈ S′(R) be the inverse Fourier transform of C_{α}^{−1}(ξ − i0)^{1+α}. Then supp E_{α} ⊂ [0, +∞[, H_{α} ∗ E_{α} = δ. When 1 + α ∈ N, then E_{α} is a constant times some derivative of the delta function, and in general, we have

E_{α}_{|}_{]0,+∞[}(x) = C̃_{α}x^{−2−α}.

We can now invert (9.3) and get

μ = \frac{1}{C_{n}} E_{\frac{n}{2} −1} ∗ ω.  (9.4)

If near some point t_{0} ∈ R, we know that μ extends to a holomorphic function, then by contour deformation in (9.3), we see that ω will have the same property. (It is easy to see how to get such a bounded extension by contour deformation, and an easy way to see that this extension is also holomorphic, is to regularize ω and E_{\frac{n}{2} −1} by convolving with Gaussians, and then let the Gaussians converge to the delta-function.) The converse implication holds by (9.4). It follows that μ and ω have the same analytic singular support. Since μ is a real distribution we know that its analytic wavefront set WF_{a}(μ) is of the form {(x, ξ) ∈ R×(R\{0}); x ∈ sing supp_{a}(μ)}. The same holds for ω and since these two distributions have the same analytic singular support, we conclude that

WF_{a}(μ) = WF_{a}(ω).  (9.5)

We refer to [73] for basic properties of the analytic wavefront set WF_{a}.

We now discuss resonances close to analytic singularities of μ on ]0, +∞[. Let P_{·} = −h^{2}Δ + V_{·}(x), where V_{·} ∈ C^{∞}(R^{n}; R), · = 0, 1, and assume that the general assumptions in Chapter 8 are verified, with Φ_{·}(\frac{1}{h^{2}} ) = h^{−n}. Then we can define μ as above.

Theorem 9.1 Let 0 < E_{0} ∈ sing supp_{a}(μ). Then for every complex neighborhood W of E_{0}, there exist h_{0} = h_{0}(W) > 0, and C = C(W) > 0, such that for h < h ≤ h_{0}, we have ∑_{0}^{1} ♯(Res (P_{·}) ∩ W) ≥ \frac{1}{C(W)}h^{−n}.

The following corollary is stronger than the theorem:

Corollary 9.2 Let P_{1} = −h^{2}Δ + V_{1}(x), V_{1} ∈ C^{∞}(R^{n}; R) satisfy all the assumptions of the local trace formula (Th. 8.2) that make sense for one single operator. Let 0 < E_{0} ∈ sing supp_{a}(ν_{+,1}). Then for every complex neighborhood W of E_{0}, there exist h_{0} = h_{0}(W) > 0, and C = C(W) > 0, such that for h < h ≤ h_{0}, we have ♯(Res (P_{1}) ∩ W) ≥ \frac{1}{C(W)}h^{−n}.

Proof of the Corollary. It suffices to construct P_{0} = −h^{2}Δ + V_{0}(x), with V_{0} ∈ C_{b}^{∞}(R^{n}; R) such that (P_{1}, P_{0}) satisfies the assumptions of the local trace formula and such that:

sup supp ν_{+,0} < E_{0},  (9.6)

∃ a complex neighborhood W_{0} of E_{0} such that  (9.7)
Res P_{0} ∩ W_{0} = ∅, when h > 0 is small enough.

We shall produce V_{0} from V_{1} by cut-off and regularization. Put K(x) = C_{n}e^{−x^{2}/2} with C_{n} > 0 chosen so that ∫_{R^{n}} K(x)dx = 1. Put K_{λ}(x) = λ^{−n}K(λ^{−1}x), for λ > 0. We make an x-dependent choice of λ:

λ(R, x) = R⟨R^{−1}x⟩^{−N_{0}},

where N_{0} > 0 is sufficiently large, depending on the dimension and R ≥ 1 is a large parameter. Put K_{R}(x, y) = K_{λ(R,x)}(x − y). Let χ ∈ C_{0}^{∞}(B(0, 2); [0, 1]) be equal to 1 on B(0, 1). For R large enough, put

V_{0}(x) = ∫ K_{R}(x, y)(1 − χ(R^{−1}y))V_{1}(y)dy.

Let θ > 0 be small but independent of R. Then in the domain |Im x| < θ⟨Re x⟩:

|V_{0}(x)| ≤ ϵ(R), ϵ(R) → 0, R → ∞,
|V_{1}(x) − V_{0}(x)| ≤ C(R)⟨x⟩^{−n−1}.

Then we have (9.6) and using that the resonances near E_{0} can be viewed as eigenvalues of P_{0}_{|}_{e^{iθ/2}R^{n}}, we see that (9.7) also holds. #

Proof of Theorem 9.1. We have seen that WF_{a}(μ) = WF_{a}(ω) and since μ, ω are real, it is clear that (E_{0}, 1), (E_{0}, −1) ∈ W F_{a}(ω). Considering the definition of WF_{a}(ω) by means of the FBI-transform ([73]), we see that there exist sequences (α_{j}, β_{j}) → (E_{0}, 1) in R^{2}, λ_{j} → +∞, ϵ_{j} ↘ 0, such that

| ∫ e^{iλ_{j}(β_{j}(α_{j}−E)+ \frac{i}{2} (α_{j}−E)^{2})}χ(E)ω(E)dE| ≥ e^{−ϵ_{j}λ_{j}},  (9.8)

where χ ∈ C_{0}^{∞}(]0, ∞[) is equal to 1 near E_{0} and has its support in a small neighborhood of E_{0}. Let a, b, a/b be small and positive and let

Ω =]E_{0} − b, E_{0} + b[+i] − a, a], W =]E_{0} − \frac{b}{2}, E_{0} + \frac{b}{2}[+i] − \frac{a}{2}, a].

Let I, J be the intersections of W, Ω with the real line and choose χ ∈ C_{0}^{∞}(J) equal to 1 near the closure of I. Let

f_{j}(E) = e^{iλ_{j}(β_{j}(α_{j}−E)+ \frac{i}{2} (α_{j}−E)^{2})}.

Then

|f_{j}_{|}_{Ω\W} | ≤ e^{− \frac{1}{C_{0}} λ_{j}},

and (9.8) reads

| ∫ (χf_{j})(E)ω(E)dE| ≥ e^{−ϵ_{j}λ_{j}}.  (9.9)

The trace formula gives

′′tr [(χf_{j})(P_{·})]_{0}^{1}′′ = [ ∑_{Res (P_{·})∩W_{−}} f_{j}(λ)]_{0}^{1} + O(h^{−n})e^{− \frac{1}{C_{0}} λ_{j}}.  (9.10)

On the other hand we have a trace formula of D. Robert [67], which in our situation implies:

tr [(χf_{j})(P_{·})]_{0}^{1} = \frac{1}{(2πh)^{n}} ∫ (χf_{j})(E)ω(E)dE + O_{j}(h^{1−n}).  (9.11)

(See the appendix of this chapter for an outline of a proof.)

Combining (9.10), (9.11), we get:

[ ∑_{Res (P_{·})∩W_{−}} f_{j}(λ)]_{0}^{1} = \frac{1}{(2πh)^{n}} ∫ (χf_{j})(E)ω(E)dE + O(h^{−n})e^{− \frac{1}{C_{0}} λ_{j}} + O_{j}(h^{1−n}),

so by (9.9) :

|[ ∑_{Res (P_{·})∩W_{−}} f_{j}(λ)]_{0}^{1}| ≥ \frac{e^{−ϵ_{j}λ_{j}} − O(1)e^{− \frac{1}{C_{0}} λ_{j}}}{(2πh)^{n}} + O_{j}(h^{1−n}).

Choose first j large enough to see that the previous expression is

≥ \frac{\frac{1}{2}e^{−ϵ_{j}λ_{j}}}{(2πh)^{n}} + O_{j}(h^{1−n}).

Then fix j and choose h small enough to see that it can be bounded from below by (2πh)^{−n}e^{−ϵ_{j}λ_{j}}/3. The theorem follows. #

We next consider resonances generated by analytic singularities on the negative axis.

Theorem 9.3 We make the same general assumptions as in Theorem 9.1 and assume that the angle of scaling θ_{0} is > π/2. Let 0 > E_{1} ∈ sing supp_{a}(μ). Let γ : [0, 1] → C be a C^{1} curve with γ(0) ∈] sup supp (μ), +∞[, γ(1) = E_{1}, Im γ(t) < 0 for 0 < t < 1. Also assume that γ is injective and γ′(t) ≠ 0, ∀t. Then for every neighborhood W of γ([0, 1]), there exist constants C, h_{0} > 0, such that

∑_{0}^{1} #(Res (P_{·}) ∩ W) ≥ \frac{1}{C} h^{−n}, 0 < h ≤ h_{0}(W).

Corollary 9.4 Let P_{1} = −h^{2}Δ + V_{1}(x) satisfy all the assumptions of the local trace formula that make sense for one of the operators. Let the angle of scaling satisfy θ_{0} > π/2, and assume that V_{1}(x) = O(⟨x⟩^{−_{n}}), for some ñ > n in the region appearing in the corresponding complex scaling assumption. Let 0 > E_{1} ∈ sing supp_{a}(ν_{−,1}), and let γ : [0, 1] → C be a C^{1} curve with γ(0) ∈ ]sup supp (ν_{+,1}), +∞[, γ(1) = E_{1}, Im γ(t) < 0 for 0 < t < 1, γ injective and γ′(t) ≠ 0 everywhere. Then for every neighborhood W of γ([0, 1]), there exist C, h_{0} > 0 such that

#(Res (P_{1}) ∩ W) ≥ \frac{1}{C} h^{−n}, 0 < h ≤ h_{0}(W).

To get the corollary, it suffices to apply Theorem 9.3 with P_{0} = −h^{2}Δ. If we combine the two corollaries, we get the following general statement:

Consider P_{1} = −h^{2}Δ + V_{1}(x) with angle of scaling θ_{0} > \frac{π}{2} and with V_{1} = O(⟨x⟩^{−_{n}}), ñ > n in the corresponding scaling region. If V_{1} ≢ 0, then #(P_{1} ∩W) ≥ \frac{1}{C(W)}h^{−n}, 0 < h ≤ h_{0}(W), where W is either any fixed complex neighborhood of some point on ]0, +∞[ or any neighborhood of a curve as in Corollary 9.4.

We refer to [77] for the proof of Theorem 9.3, and we here only explain some of the ideas. Let γ, W be as in the theorem and assume that W is open in {z ∈ C \ {0}; −2θ_{0} < arg z ≤ ϵ_{0}} for some small ϵ_{0} > 0 (as in the local trace formula). We may assume that the intersection of W with the real line is the union of two intervals I_{±} ⊂ R_{±}, where R_{±} denote the open half-axes. Let Ω ⊃⊃ W have the same properties as W and let J_{±} be the corresponding intervals. Let χ̂ ∈ C_{0}^{∞}(Ω) be equal to 1 on \overline{W}, almost holomorphic at J_{±}. For a suitable sequence f = f_{j} (that will not be constructed in detail here), with |f| ≤ 1 on Ω \ W, we write the trace formula:

[ ∑_{λ∈Res ∩W_{−}} f_{j}(λ)]_{0}^{1} = ′′tr [χ_{+}f_{j}(P_{·})]_{0}^{1}′′ +′′tr [χ_{−}f_{j}(P_{·})]_{0}^{1}′′ + O(h^{−n}).

Using the semiclassical trace formula, we get:

′′tr [χ_{±}f_{j}(P_{·})]_{0}^{1}′′ = \frac{1}{(2πh)^{n}} ∫ (χ_{±}f_{j})(E)ω(E)dE + O_{j}(h^{1−n}).

ω(E) extends analytically from J_{+} to a function ω_{+} on Ω ∩ {Im z < 0} and by Stokes’ formula, we get

∫ χ_{+}f_{j}(E)dE = − ∫ (χ_{−}f_{j})(E)ω_{+}(E − i0)dE + O(1).

It follows that

[ ∑_{λ∈Res P_{·}∩W_{−}} f_{j}(λ)]_{0}^{1} =  (9.12)

\frac{1}{(2πh)^{n}} ∫ (χ_{−}f_{j})(E)(ω(E) − ω_{+}(E − i0))dE + O(h^{−n}) + O_{j}(h^{1−n}).

Here ω(E) − ω_{+}(E − i0) can be obtained from μ by convolution and we can show that (E_{1}, ±1) both belong to the analytic wavefront set of ω(E) − ω_{+}(E − i0).

Approximating Gaussian functions by suitable functions f_{j} as above, we can arrange so that the first term in the right hand side of (9.12) is ≫ O(h^{−n}) which permits us to conclude. #

We shall next consider clouds of resonances for systems, following the the article of L. Nedelec [59]. Schrödinger operators with matrix-valued potentials and similar systems appear frequently in quantum mechanics, for instance when one reduces the dimension by means of the Born-Oppenheimer reduction and similar methods that lead to a so called effective Hamiltonian. Although these effective Hamiltonians are not in general exactly equal to Schrödinger operators with matrix-valued potential, the latter form important models for more general systems that could also be studied with more or less the same methods.

Let

P = −h^{2}Δ + V (x),  (9.13)

with V (x) ∈ C_{b}^{∞}(R^{n}; Hermitian r × r matrices ), satifying (H1) ∃θ_{0} ∈ [0, \frac{π}{2}[, ϵ > 0, R > 0, such that V extends to a holomorphic function on {rω; ω ∈ C^{n}, dist (ω, S^{n−1}) < ϵ, r ∈ e^{i[0,θ_{0}]}]R, +∞[}.

(H2) There exists a Hermitian r × r-matrix V_{0} and a number ñ > n such that ∥V (x) − V_{0}∥ ≤ C⟨x⟩^{−_{n}} in the set in (H1).

Put P_{0} = −h^{2}Δ + V_{0}, and let λ_{1,0} ≤, .. ≤ λ_{r,0} be the eigenvalues of V_{0}, x ∈ R^{n}. As in the scalar case, we can then define the resonances of P as the eigenvalues of the scaled operator P_{θ} = P_{|}_{Γ_{θ}}, 0 < θ ≤ θ_{0} in the set R \ ∪_{j=1}^{r}(λ_{j,0} + e^{−2iθ}]0, +∞[), and the resonances are independent of θ in the sence that for 0 < θ_{1} < θ_{2} ≤ θ_{0}, θ_{1} ≤ θ ≤ θ_{2}, and for every 1 ≤ j ≤ r − 1, the eigenvalues of P_{θ} in the triangle bounded by the interval ]λ_{j,0}, λ_{j+1,0}[ and the half-rays λ_{j,0} + e^{−2iθ_{1}}]0, +∞[, λ_{j+1,0} + e^{−2iθ_{2}}]0, +∞[ are independent of θ. A similar statement holds in the sector λ_{r,0} + e^{−2i[0,θ_{0}[}]0, +∞[.

Assume λ_{j,0} < λ_{j+1,0} for some 1 ≤ j ≤ r and let W ⊂⊂ Ω be open and relatively compact in

(λ_{j,0} + e^{i]−2θ_{0},ϵ_{0}]}]0, ∞[) ∩ (]λ_{j,0}, λ_{j+1,0}[+e^{−2iθ_{0}}R).

Here ϵ_{0} > 0 is small and we use the convention that λ_{r+1,0} = +∞. Assume that I := W ∩ R, J := Ω ∩ R are intervals.

The local trace formula still holds:

Theorem 9.5 Let f = f(z; h) be holomorphic for z ∈ Ω with |f(z; h)| ≤ 1, z ∈ Ω \ W. Let 1_{I} ≺ χ ∈ C_{0}^{∞}(J) be independent of h. Let P_{1} := P. Then [χf(P_{·})]_{0}^{1} is of trace class and

tr [χf(P_{·})]_{0}^{∞} = ∑_{z∈Res (P)∩W_{−}} f(z; h) + O(h^{−n}),

where W_{−} = W ∩ e^{−2i[0,θ_{0}[}]0, +∞[

The proof (see [59]) is essentially the same as in the scalar case.

On the other hand, we have Robert’s trace formula:

Theorem 9.6 Let g ∈ C_{0}^{∞}(R) be independent of h. Then [g(P_{·})]_{0}^{1} is of trace class and

tr [g(P_{·})]_{0}^{1} = \frac{1}{(2πh)^{n}} ∫ ∫ tr (g(p(x, ξ)) − g(p_{0}(x, ξ)))dxdξ + O(h^{1−n}),

where (x, ξ) = ξ^{2} + V (x), p_{0}(x, ξ) = ξ^{2} + V_{0}(x).

Here the integral can also be written

∑_{j} ∫ ∫ (g(ξ^{2} + λ_{j}(x)) − g(ξ^{2} + λ_{j,0}(x)))dxdξ =: ⟨ω, g⟩,

where λ_{1}(x) ≤ ... ≤ λ_{r}(x) denote the eigenvalues of V (x). As in the scalar case,

ω = C_{n}(E)_{+}^{\frac{n}{2} −1} ∗ μ, where ⟨μ, ϕ⟩ = ∑_{j} ∫ (ϕ(λ_{j}(x)) − ϕ(λ_{j,0}))dx.

For E ∈]λ_{j,0}, λ_{j+1,0}[ we put

ν_{j}(E) = − ∑_{k=1}^{j} ∫_{λ_{k}(x)≥E} dx + ∑_{k=j+1}^{r} ∫_{λ_{k}(x)≤E} dx

Then on this interval, we have

μ = \frac{d}{dE} ν_{j}, sing supp_{a}(ω) = sing supp_{a}(μ) = sing supp_{a}(ν_{j}).

The arguments in the scalar case extend (see [59]) and give

Theorem 9.7 Let ]λ_{j,0}, λ_{j+1,0}[∋ E_{0} ∈ sing supp_{a}(ν_{j}). Then for any neighborhood W of E_{0}, there exist h_{0}, C > 0, such that

#(Res (P) ∩ W) ≥ \frac{1}{C} h^{−n}, 0 < h ≤ h_{0}.

If λ_{k}(x) is a simple eigenvalue near some point x_{0}, then λ_{k}(x) is a smooth function there, and if E_{0} = λ_{k}(x_{0}) is a critical value, we expect in general that E_{0} will belong to the analytic singular support of μ. This is essentially as in the scalar case. Multiple eigenvalues can also give rise to analytic singularities of μ and we now describe such a situation:

Let E_{0} ∈]λ_{j,0}, λ_{j+1,0}[, 1 ≤ j ≤ r, with the convention λ_{r+1,0} = +∞, and assume:

0) n = 2, V is real and symmetric.

1) For x = 0, there are exactly two eigenvalues λ_{k}(0), λ_{k+1}(0) that are equal to E_{0}.

2) If E_{0} = λ_{ν}(x) for some ν and some x ≠ 0, then λ_{ν}(x) is a simple eigenvalue, and dλ_{ν}(x) ≠ 0.

3) Let a(x)I + ( \begin{matrix}c(x) & b(x) \\ b(x) & −c(x) \end{matrix} ) be a smooth representative of the restriction

of V (x) to the 2-dimensional eigenspace corresponding to the two eigenvalues λ_{k}(x), λ_{k+1}(x), for x in a neighborhood of 0. Then dc(0), db(0) are linearly independent and da(0) = βdb(0) + γdc(0) with β^{2} + γ^{2} < 1. Under these assumtions it is proved in [59] that E_{0} ∈ Sing supp_{a}μ.

Appendix.

We first recall the operator version of the Cauchy-Riemann-Green-Stokes formula, which was used (in a slightly different operator version) by Dynkin [22] and has been used very much since the work [39]. Let P : H → H be some unbounded selfadjoint operator, and let f ∈ C_{0}^{∞}(R). Let f̃ ∈ C_{0}^{∞}(C) be an almost holomorphic extension of f with the property that \overline{∂}f̃ vanishes to infinite order on the real axis). Then since 1/(πz) is a fundamental solution of the Cauchy-Riemann operator \overline{∂}, we have

f(E) = −\frac{1}{π} ∫ \overline{∂}f̃(z)\frac{1}{z − E} L(dz), E ∈ R.  (A.1))

The operator version follows by replacing “E” by “P ”:

f(P) = −\frac{1}{π} ∫ \overline{∂}f̃(z)(z − P)^{−1}L(dz), E ∈ R.  (A.2))

The proof is a straight forward application of the spectral theorem. ([39], [21], [20]).

Let P_{·} be as in the main text of this chapter. Let χ ∈ C_{0}^{∞}(R) be conveniently chosen, so that the infimum of the spectrum of P̃_{.} := −h^{2}Δ + V_{.}(x) + χ(hD) is ≥ 1 + sup supp f. The resolvent identity gives

(z − P_{.})^{−1} = (z − P̃_{.})^{−1} + (z − P̃_{.})^{−1}χ(hD)(z − P_{·})^{−1}.  (A.3)

Using this in (A.2) we get, since f(P̃_{·}) = 0:

f(P_{·}) = −\frac{1}{π} ∫ \overline{∂}f̃(z)(z − P̃_{.})^{−1}χ(hD)(z − P_{·})^{−1}L(dz).  (A.4)

As for instance in the proof of the invariance of the condition (7.12), we see that [(z − P̃_{·})^{−1}χ(hD)(z − P_{.})^{−1}]_{0}^{1} is of trace class with corresponding norm ≤ O(1)h^{−n}|Im z|^{−N_{0}}, for z ∈ neigh supp (f̃). A convenient way to see this, is to put P_{t} = (1 − t)P_{0} + tP_{1}, and to write

[(z − P̃_{·})^{−1}χ(hD)(z − P_{·})^{−1}]_{0}^{1} = ∫_{0}^{1} \frac{∂}{∂t}((z − P̃_{t})^{−1}χ(hD)(z − P_{t})^{−1})dt, (A.5)

then develop the right hand side and use the same arguments as in Chapter 7.

It follows that for any fixed δ > 0:

tr [f(P_{.})]_{0}^{1} = −\frac{1}{π} tr ∫_{|Im z|>h^{δ}} \overline{∂}f̃(z)[(z −P̃_{.})^{−1}χ(hD)(z −P_{·})^{−1}]_{0}^{1}L(dz)+O(h^{∞}).

(A.6)

Let 0 < δ < \frac{1}{2}. Then (z − P_{t})^{−1} is an h-pseudodifferential operator with symbol a of class S_{δ}(h^{−δ}⟨ξ⟩^{−2}) for |Im z| > h^{δ}, in the sense that

∂_{x}^{α}∂_{ξ}^{β}a = O(h^{−δ(|α|+|β|)}h^{−δ}⟨ξ⟩^{−2}).

(The detailed proof of this uses a semi-classical version of a lemma of R. Beals, see [39] and also [21].) In this class, the symbol has an asymptotic expansion:

∼ \frac{1}{(z − p_{t}(x, ξ))} + \frac{ha_{1}(x, ξ, z)}{(z − p_{t})^{3}} + \frac{h^{2}a_{2}(x, ξ, z)}{(z − p_{t})^{5}} + ..., .

[[Nästa version, skriv lite mer om egenskaperna hos a_{j}.]] (See [21].) Using this in (A.5), we see that [(z − P̃_{·})^{−1}χ(hD)(z − P_{·})^{−1}]_{0}^{1} has a symbol of class S_{δ}(h^{−2δ}⟨ξ⟩^{−N}⟨x⟩^{−_{n}}) for every n > 0. The symbol is equal to

[(z − p̃_{·})^{−1}χ(ξ)(z − p_{·})^{−1}]_{0}^{1} + ∫ O(h)⟨x⟩^{−_{n}}⟨ξ⟩^{−N}|z − p_{t}|^{−N_{0}}dt.  (A.7)

We plug this into (A.6) and use that an h-pseudodifferential operator with symbol a in S_{δ}(m) is of trace class if m is integrable, and has the trace \frac{1}{(2πh)^{n}} ∫ ∫ a(x, ξ)dxdξ. It follows that

tr [f(P_{·})]_{0}^{1} = −\frac{1}{π} ∫ ∫ ∫ \overline{∂}f̃(z)[(z−p̃_{·})^{−1}χ(ξ)(z−p_{·})^{−1}]_{0}^{1} \frac{dxdξ}{(2πh)^{n}} L(dz)+O(h^{1−n}).

(A.8)

Here

−\frac{1}{π} ∫ \overline{∂}f̃(z)[(z − p̃_{·}(x, ξ))^{−1}χ(ξ)(z − p_{·})]_{0}^{1}L(dx) =

−\frac{1}{π} ∫ \overline{∂}f̃(z)[(z − p_{·})^{−1} − (z − p̃_{·})^{−1}]_{0}^{1}L(dx) =

[f(p_{.}(x, ξ)) − f(p̃_{.}(x, ξ))]_{0}^{1} = [f(p_{·}(x, ξ))]_{0}^{1},

since f(p̃_{·}(x, ξ)) = 0 by construction.

It follows that

tr [f(P_{·})]_{0}^{1} = \frac{1}{(2πh)^{n}} ∫ ∫ [f(p_{·}(x, ξ))]_{0}^{1}dxdξ + O(h^{1−n}),

as claimed.

10 Resonances generated by closed trajectories.

In this chapter we show how closed classical trajectories can give rise to singularities in the local trace-formula, leading to the existence of many resonances. Here enters also some version of the so called Gutzwiller trace formula. This strategy was used by Bardos-Lebeu-Rauch [7], in order to show that there are infinitely many resonances associated to certain types of strictly convex obstacles, and later by Ikawa [45] in the case of an obstacle composed by several strictly convex disjoint parts. Quantitative results with lower bounds on some counting function for the resonances were obtained by Sjöstrand and Zworski [84]. The lower bounds in [84] were originally obtained by using the exact Bardos-Guillot-Ralston trace-formula, and hence limited to the case of odd space-dimensions, however the local trace-formula can be used here as well ([75]), and as a consequence we avoid the restriction on the dimension. Here we prefer to discuss an analogous recent result by J.F. Bony [11], which treats explicitly the semi-classical situation. It is also a bit sharper than the corresponding result in [84].

Let

P = ∑_{|α|≤2} a_{α}(x; h)(hD_{x})^{α} : L^{2}(R^{n}) → L^{2}(R^{n})  (10.1)

satisfy the general assumptions of Chapter 7. Assume for simplicity that

a_{α}(x; h) ∼ ∑_{j≥0} a_{α}^{j}(x)h^{j}  (10.2)

in C_{b}^{∞} and that this also holds in the complex domain (as in (7.41)). Let

p = p_{0}(x, ξ) = ∑_{|α|≤2} a_{α}^{0}(x)ξ^{α}  (10.3)

be the semi-classical principal symbol of P. Let

H_{p} = ∑_{j=1}^{n}(\frac{∂p}{∂ξ_{j}} \frac{∂}{∂x_{j}} − \frac{∂p}{∂x_{j}} \frac{∂}{∂ξ_{j}})  (10.4)

be the corresponding Hamilton field.

Let E_{0} > 0, T_{0} > 0 and assume that γ : R ∋ t ↦ γ(t) ∈ p^{−1}(E_{0}) is a T_{0}-periodic H_{p}-trajectory along which dp ≠ 0. Let us recall some properties of the associated Poincaré map: At a point ρ_{0} = γ(t_{0}), the space T_{ρ_{0}}(p^{−1}(E_{0}))/RH_{p}(ρ_{0}) is a symplectic vectorspace: If σ = ∑ dξ_{j} ∧ dx_{j} is the standard symplectic form that we also view as a bilinear form: ⟨σ, t ∧ s⟩ = t_{ξ} · s_{x} − t_{x} · s_{ξ}, t = (t_{x}, t_{ξ}), s = (s_{x}, s_{ξ}), then the symplectic orthogonal space of T_{ρ_{0}}(p^{−1}(E_{0})) is RH_{p} ⊂ T_{ρ_{0}}(p^{−1}(E_{0})) and we get a natural non-degenerate alternate bilinear form on T_{ρ_{0}}(p^{−1}(E_{0}))/RH_{p}(ρ_{0}). A more or less equivalent version of this, is that if Σ ⊂ p^{−1}(E_{0}) is a hypersurface (i.e. of dimension (2n − 1) − 1 = 2(n − 1)) passing through ρ_{0} transversally to H_{p}(ρ_{0}), then near ρ_{0}, (Σ, σ_{|}_{Σ}) is a symplectic manifold.

The (non-linear) Poincaré map is then obtained in the following way: Start at ρ ∈ Σ∩neigh (ρ_{0}) and follow the H_{p}-trajectory until you hit Σ again for a unique time t(ρ) = T_{0} +O(|ρ−ρ_{0}|), at the point κ(ρ) = exp(t(ρ)H_{p})(ρ). κ is then the non-linear Poincaré map. (If ρ̃_{0} = γ(s_{0}) is a different base point, and Σ̃ ⊂ p^{−1}(E_{0}) a hypersurface transversal to H_{p}(ρ̃_{0}), let κ̃ be the corresponding Poincaré map. Then

κ = α^{−1} ∘ κ̃ ∘ α,

where α : Σ ∩ neigh (ρ_{0}) → Σ̃ ∩ neigh (ρ̃_{0}) is given by α(ρ) = exp(r(ρ)H_{p}(ρ), with t(ρ) = s_{0} − t_{0} + O(|ρ − ρ_{0}|), . Then it is easy to check that the non-linear Poincaré map κ : Σ ∩ neigh (ρ_{0}) → Σ ∩ neigh (ρ_{0}) is a canonical transformation (or symplectomorphism, i.e. a map which conserves the symplectic form) and the corresponding linearized Poincaré map dκ(ρ_{0}) : T_{ρ_{0}}Σ → T_{ρ_{0}}Σ will of course be a symplectomorphism also. Notice that T_{ρ_{0}}Σ ≃ T_{ρ_{0}}(p^{−1}(E_{0}))/(RH_{p}(ρ_{0})). Since dκ(ρ_{0}) is a real symplectomorphism, we know that its eigenvalues can be grouped into the following families:

1) The value 1 with even multiplicity,

2) The value –1 with even multiplicity.

3) The values λ and λ^{−1} for some λ ∈] − ∞, −1[∪]1, +∞[

4) The values λ, \overline{λ} for some λ with |λ| = 1, Im λ > 0.

5) The values λ, λ^{−1}, \overline{λ}, \overline{λ}^{−1}, for some λ ∈ C with ℑλ > 0, |λ| > 1. For each of the groups in 3)–5), the eigenvalues have the same multiplictity.

Assume that the (linearized) Poincaré map is non-degenerate in the sense that 1 is not an eigenvalue of dκ(ρ_{0}). Then by the implicit function theorem, applied to the map ρ ↦ κ(ρ) − ρ, we see that ρ_{0} is an isolated fixed point of κ and this situation is stable under small perturbations of p − E_{0}. We can therefore vary E ∈ neigh (E_{0}, R) and see that for each such energy, there is a unique closed trajectory γ_{E} in p^{−1}(E) of period T (E) = T_{0} + O(E − E_{0}). Moreover this trajectory depends smoothly on E in the obvious sense.

The set S = ∪_{E∈neigh (E_{0},R)}R(γ_{E}) (where “R”) stands for “range of”, is a smooth symplectic 2-dimensional manifold. Write P = dκ_{E}(ρ_{E}), where ρ_{E} ∈ R(γ_{E}) and κ_{E} denote a fixed point and Poincaré map at energy E.

Assume that γ_{0} = γ_{E_{0}} is (up to trivial reparametrizations) the only T_{0} periodic H_{p}-trajectory in p^{−1}(E_{0}). (This assumption can be weakened to allow for finitely many trajectories, obtained from γ_{0} by the action of some finite symmetry group.) Notice that this assumption implies that E_{0} is not a critical value of p.

Under the above assumptions we have the following theorem, due to J.F. Bony [11].

Theorem 10.1 Let ϵ, μ > 0. Then ∃ C > 0 such that for every interval [a(h), b(h)] ⊂ [E_{0} − \frac{1}{C} , E_{0} + \frac{1}{C}], 0 < h ≤ \frac{1}{C}, we have

∑_{λ∈Res P ∩W} e^{T (Re λ)(Im λ)(1−μ)/h} ≥  (10.5)

\frac{1}{2πh} ∫_{a(h)}^{b(h)} T^{∗}(E)| det(1 − P_{γ_{E}})|^{−1/2}dE − O(h^{1/C})\frac{b(h) − a(h)}{h},

where

W = \{λ ∈ C; a(h) − Ch log \frac{1}{h} ≤ Re λ ≤ b(h) + Ch log \frac{1}{h},

−\frac{h log \frac{1}{h}}{T (Re λ)}(n − 1 + ϵ\frac{log(b(h) − a(h))}{log h}) ≤ Im λ ≤ 0\}

and T^{∗}(E) > 0, is the primitive period, i.e. the smallest positive period of γ_{E}.

Corollary 10.2 Under the same assumptions, we have the same lower bound on #(Res (P) ∩ W).

Outline of the proof. Let P_{j} be two operators as above, with [a_{α,j}]_{0}^{1} = O(1)⟨x⟩^{−_{n}}, for some ñ > n. We have then an extension of the local trace formula: Let

Ω =] − b, b[+i] − a, a], W =] − \frac{b}{2}, \frac{b}{2}[+i] − \frac{a}{2}, a], a, b > 0.

Let I, J be the intersections of W, Ω with the real line and choose χ ∈ C_{0}^{∞}(J) equal to 1 near the closure of I. Put Ω_{δ} = E_{0} + δΩ, W_{δ} = E_{0} + δW, χ_{δ}(E) = χ(\frac{E−E_{0}}{δ}).

Theorem 10.3 Let Ch < δ < 1/C, with C ≫ 1 sufficiently large. Let f(z; h) be holomorphic in Ω_{δ} with |f(z; h)| ≤ 1 for z ∈ Ω_{δ} \ W_{δ}. Then

tr [(χ_{δ}f)(P_{·}; h)]_{0}^{1} = [ ∑_{λ∈Res P} f(λ; h)]_{0}^{1} + O(1)δ^{−1}h^{−n}.  (10.6)

The proof is an adaptation of the usual one. See [11]. Another basic ingredient is the Gutzwiller trace formula, which in this version was obtained by D. Robert [67] and to which J.F. Bony [11] has given a detailed proof, involving the approximation of e^{−itP_{·}/h} by Fourier integral operators.

We now describe that formula in the special case that we will need it. Let A > 0 be small and let g(t) ∈ C_{0}^{∞}(neigh (T ([E_{0} − A, E_{0} + A]); R)) be equal to 1 near T ([E_{0} − A, E_{0} + A]). Put

f_{g}(F) = f_{g,C,E}(F) = ∫ e^{−it(F −E)/h}g(t)e^{−(t−T (E))^{2}C(log h^{−1})/2}dt.

Let

f(F) = f_{1}(F) = \begin{matrix}√ & \overline{2π} \\ √ & \overline{C log h^{−1}} \end{matrix} e^{−iT (E)(F −E)/h− \frac{(F −E)^{2}}{2Ch^{2} log(h^{−1})}} .

Let 1_{[E_{0}−2A,E_{0}+2A]} ≺ χ ∈ C_{0}^{∞}(]E_{0} − 3A, E_{0} + 3A[), and put P_{1} = P. It is possible to construct P_{0} as in the local trace-formula, so that P_{0} has no resonances in a fixed h-independent neighborhood of [E_{0} −A, E_{0} +A], and so that the principal symbol p_{0} has no closed classical trajectories with energy in [E_{0} − 3A, E_{0} + 3A] with period in some fixed neighborhood of T ([E_{0} − 3A, E_{0} + 3A])

Theorem 10.4 For E ∈ [E_{0} − A, E_{0} + A] and h small enough, we have

tr [(χ^{2}f_{g})(P_{j})]_{0}^{1} =
e^{iS(γ_{E})/h+i(σ(γ_{E})π/4−S_{1}(γ_{E}))}T^{∗}(E)| det(1 − P_{γ_{E}})|^{− \frac{1}{2}} + O(h log h^{−1}).

Here S(γ_{E}) = ∫_{γ_{E}} ξ · dx is the action of γ_{E} and σ(γ_{E}) and S_{1}(γ_{E}) are realvalued.

We can now end the outline of the proof of Theorem 10.1. Combining the last two theorems we get

∑_{λ∈Res (P)∩W_{E,h}} |f_{g,E}(λ)| ≥ T^{∗}(E)| det(1 − P_{γ_{E}})|^{−1/2} − O(h log \frac{1}{h}).

[[Här blir mitt handskrivna manus alltför otydligt. Skall återkomma till detta senare! Beviset avslutas i alla fall med att man integrerar olikheten m.a.p. E.]]

11 From quasi-modes to resonances.

We shall describe a result of P. Stefanov which culminates a series of works, started by Stefanov–Vodev [88] and continued by Tang–Zworski [89]. Actually many of the basic ideas can be traced back much further to works of Agmon, Agranovich and others in connection with the completeness of the set of generalized eigenfunctions for non-selfadjoint operators.

Let P = P (h) : H → H be a a semiclassical black box operator as in chapter 7, for which we can define the resonances in e^{i]−2θ_{0},0]}]0, +∞[. As already mentioned in that chapter, we can also define the resonances as the poles of the meromorphic extension from the upper half-plane across ]0, +∞[ of (z − P)^{−1} : H_{comp} → D_{loc}.

We need some remarks about truncated spectral projections. Let

1_{B(0,R_{0})} ≺ χ ∈ C_{0}^{∞}(R^{n})

be such that “the exterior” operator Q has analytic coefficients in a neighborhood of supp (1 − χ). We choose Γ_{θ} so that Γ_{θ} = R^{n} near supp χ. Let z_{0} ∈ e^{i]−2θ,0]}]0. + ∞[ be a resonance, 0 < θ ≤ θ_{0}. Let

π_{θ,z_{0}} = \frac{1}{2πi} ∫_{∂D(z_{0},ϵ)}(z − P_{θ})^{−1}dz, P_{θ} = P_{|}_{Γ_{θ}}, 0 < ϵ ≪ 1,  (11.1)

be the corresponding spectral projection. We know that rank π_{θ,z_{0}} is independent of the choice of θ with −\frac{1}{2}arg z_{0} < θ ≤ θ_{0}.

Lemma 11.1 rank (χπ_{θ,z_{0}}) = rank (π_{θ,z_{0}}).

Proof. We know that R(π_{θ,z_{0}}) = N ((P_{θ} − z_{0})^{N_{0}}) for some 1 ≤ N_{0} ≤ m(z_{0}). Let u ∈ R(π_{θ,z_{0}}) with χu = 0. From the extension lemma 7.2, we conclude that u = 0. It follows that rank (π_{θ,z_{0}}) ≤ rank (χπ_{θ,z_{0}}), and the lemma follows since the opposite inequality is obvious. #

Recall that H_{θ} = H_{R_{0}} ⊕ L^{2}(Γ_{θ} \ B(0, R_{0})). Let Γ_{θ} = \overline{Γ}_{θ}. If u ∈ L^{2}(Γ_{θ}), then u^{∗}(x) := \overline{u(\overline{x})} ∈ L^{2}(Γ_{−θ}), and we see that L^{2}(Γ_{−θ}) and L^{2}(Γ_{θ}) become mutually dual spaces for the scalar product (u|v) = ∫_{Γ_{θ}} u(x)v^{∗}(x)dx. This extends in the natural way to a duality between H_{θ} and H_{−θ}, and P_{−θ} : H_{θ} → H_{−θ} becomes the adjoint of P_{θ}. If z_{0} is an eigenvalue of P_{θ}, then \overline{z}_{0} is an eigenvalue of P_{−θ}, and we get π_{−θ,\overline{z}_{0}} = π_{θ,z_{0}}^{∗} for the corresponding spectral projections. It follows that

rank (π_{θ,z_{0}}χ) = rank (\overline{χ}π_{−θ,\overline{z}_{0}}) = rank (π_{−θ,\overline{z}_{0}}) = rank (π_{θ,z_{0}}).

Hence R(π_{θ,z_{0}}) = R(π_{θ,z_{0}}χ), and we get:

rank (π_{θ,z_{0}}) = rank (χπ_{θ,z_{0}}) = rank (π_{θ,z_{0}}χ) = rank (χπ_{θ,z_{0}}χ).  (11.2)

Let R(z) : H_{comp} → D_{loc} denote the meromorphic extension of (z − P)^{−1} from e^{i]0,ϵ_{0}[}]0, +∞[ to e^{i]−2θ_{0},ϵ_{0}[}]0, +∞[. From Chapter 7 it follows that if z_{0} is a resonance, then

rank π_{z_{0}} ≤ rank π_{θ,z_{0}}, −\frac{1}{2}arg z_{0} < θ ≤ θ_{0},  (11.3)

where

π_{z_{0}} = \frac{1}{2πi} ∫_{∂D(z_{0},ϵ)} R(z)dz.

On the other hand, since χ(z − P_{θ})^{−1}χ = χR(z)χ, it is clear that

χπ_{z_{0}}χ = χπ_{θ,z_{0}}χ,  (11.4)

so from (11.2), (11.3), we get

rank (π_{z_{0}}) = rank (χπ_{z_{0}}χ) = rank (π_{θ,z_{0}}) = rank (χπ_{θ,z_{0}}χ).  (11.5)

Recall that among our assumptions we have (7.12):

N(P^{♯}, [−λ, λ]) = O(1)Φ(\frac{λ}{h^{2}} ), λ ≥ 1.

We also made assumptions on Φ which imply that Φ(t) = O(t^{n^{♯}/2}) for some n^{♯} ≥ n. The following result is a simple long-range extension of a result of P. Stefanov [87]. (The main novelty of Stefonov’s result compared to that of Tang-Zworski [89] was that Stefanov is able to treat multiplicities in the case when quasi-modes are very close to each other.)

Theorem 11.2 Let H ⊂]0, h_{0}] have 0 as an accumulation point. Let 0 < a_{0} ≤ a(h) ≤ b(h) ≤ b_{0} < ∞ be two functions on H. Assume that for every h ∈ H, ∃ m(h) ∈ {1, 2, ...} and E_{j}(h) ∈ [a(h), b(h)], u_{j}(h) ∈ D, 1 ≤ j ≤ m(h), such that supp u_{j} ∈ K ⊂⊂ R^{n}, where K is independent of h,

∥(P (h) − E_{j}(h)))u_{j}(h)∥_{H} ≤ R(h), 1 ≤ j ≤ m(h),
|(u_{i}(h)|u_{j}(h)) − δ_{i,j}| ≤ R(h),

where R(h) = O(h^{∞}). Let S(h) be a function with

max(h^{−n^{♯}−1}R(h), e^{−D/h}) ≤ S(h) = O(h^{∞}),

for some fixed constant D > 0.

Then for every k ∈ N, ∃h(S, k) > 0, such that for H ∋ h < h(S, k), P (h) has at least m(h) resonances in

[a(h) − 6h^{k}, b(h) + 6h^{k}] + i[0, 2S(h)h^{−1−n^{♯}}].

The difficulty is that the problem is non-selfadjoint. Under the same assumptions, it is much easier to prove that P^{♯}(h) has at least m(h) eigenvalues in [a(h) − h^{k}, b(h) + h^{k}]. In particular, we know that m(h) = O(h^{−n^{♯}}). The following lemma is essentially from [89].

Lemma 11.3 For z in some fixed compact subset of e^{i]−2θ_{0},0]}]0, ∞[, there exists A > 0 such that we have

∥χ(z − P (h))^{−1}χ∥_{H→H} ≤ Ae^{Ah^{−n^{♯}} log g(h)^{−1}},

if dist (z, Res P (h)) ≥ g(h) > 0. (Here (z−P (h))^{−1} denotes the meromorphic extension of the resolvent, when Im z ≤ 0.)

Proof. Recall from Chapter 8 that

z − P_{θ} = (1 + K̃(z))(z − P̂_{θ}), θ = θ_{0},

where ∥K̃∥ = O(1). Also recall that χ(z − P (h))^{−1}χ = χ(z − P_{θ})^{−1}χ. It is therefore enough to show that

∥(1 + K̃(z))^{−1}∥ ≤ Ae^{Ah^{−n^{♯}} log \frac{1}{g}} , when dist (z, Res P) ≥ g(h).  (11.6)

Recall that

det(1 + K̃(z)) = GD_{w},

where |G| ≥ A^{−1}e^{−Ah^{−n^{♯}}} It is easy to see that |D_{w}| ≥ A^{−1}e^{−Ah^{−n^{♯}} log \frac{1}{g}}, and we get (11.6) with a new constant A. #

Notice that ∥χ(z − P)^{−1}χ∥ ≤ 1/|Im z|, for Im z > 0. We state the following consequence of the maximum principle essentially from [89].

Lemma 11.4 For h ∈ H, let F (z; h) be holomorphic near Ω_{5}(h) = [E(h) − 5h^{k}, E(h) + 5h^{k}] + i[−S(h)h^{−1−n^{♯}}, S(h)], with E(h) ∈ R, S(h) as in the theorem. Assume that

|F (z; h)| ≤ A exp(Ah^{−n^{♯}} log \frac{1}{hS(h)}) on Ω_{5}(h),

|F (z; h)| ≤ \frac{1}{|Im z|} on Ω_{5}(h) ∩ \{z; Im z > 0\}.

Then there exist h_{1} = h_{1}(S, A, k) > 0, B = B(S, A, k) > 0 (independent of h) such that

|F (z; h)| ≤ \frac{B}{S(h)}, z ∈ [E(h) − h^{k}, E(h) + h^{k}], h ∈ H, h ≤ h_{1}.

Proof. This is a routine argument for subharmonic functions. Consider the subharmonic function

f(z) = log |F (z; h)| − log (\frac{1}{S(h)})\frac{Im z + S(h)h^{−1−n^{♯}}}{S(h) + S(h)h^{−1−n^{♯}}}

−((log A) + Ah^{−n^{♯}} log (\frac{1}{hS(h)}))\frac{−Im z + S(h))}{(S(h) + S(h)h^{−1−n^{♯}}} ,

which is ≤ 0 on the horizontal segments of ∂Ω_{5}(h) and satisfies

|f(z)| ≤ (log A) + Ah^{−n^{♯}} log (\frac{1}{hS(h)})

on the vertical segments.

After the change of coordinates, z = (S + Sh^{−1−n^{♯}})w, Ω_{5} becomes a rectangle of vertical “width” 1 and of horizontal “length” \frac{10h^{k}}{S(1+h^{−1−n^{♯}})}. In such a long rectangle, the Poisson kernel decays exponentially (see for instance [76]), and it follows that in the rectangle of the same center and same width and of length \frac{2h^{k}}{S(1+h^{−1−n^{♯}})}, we have

|f(z)| ≤ ((log A) + Ah^{−n^{♯}} log (\frac{1}{hS}))e^{−h^{k}/(S(1+h^{−1−n^{♯}}))}

≤ O(h^{∞}) log (\frac{1}{hS}) ≤ log \frac{1}{S} .

Restricting this further to real z, we get

log |F (z; h)| ≤ O(1) log \frac{1}{S} ,

and the lemma follows. #

Let z_{0} be a resonance. Then we know that for z ∈ neigh (z_{0}):

(z − P_{θ})^{−1} = ∑_{j=1}^{N} \frac{π_{z_{0},θ}^{(j)}}{(z − z_{0})^{j}} + hol (z),

where

π_{z_{0},θ}^{(1)} = π_{z_{0},θ}

is the spectral projection and R(π_{z_{0},θ}^{(j)}) ⊂ R(π_{z_{0},θ}) = R(π_{z_{0},θ}χ), j ≥ 1, if we choose χ as in the beginning of this chapter. It follows that R(χπ_{z_{0},θ}^{(j)}χ) ⊂ R(χπ_{z_{0},θ}χ), and recalling that χ(z − P)^{−1}χ = χ(z − P_{θ})^{−1}χ, we get

χ(z − P)^{−1}χ = ∑_{j=1}^{N}(z − z_{0})^{−j}A_{j} + hol (z), z ∈ neigh (z_{0}),  (11.7)

with A_{j} = χπ_{z_{0},θ}^{(j)}χ, R(A_{j}) ⊂ R(A_{1}), rank (A_{1}) = m(z_{0}).

End of the proof of the theorem. Let z_{1}(h), ..., z_{M(h)}(h), be the distinct resonances of P (h) in

Ω_{6}(h) := [a(h) − 6h^{k}, b(h) + 6h^{k}] + i[−2S(h)h^{−1−n^{♯}}, S(h)],

and let

A^{(j)}(h) = A_{1,z_{j}}(h) = \frac{1}{2πi} ∫_{∂D(z_{j},ϵ)} χ(z − P)^{−1}χdz.

Here we choose χ as above, with the additional property that 1_{K} ≺ χ. Let π(h) be the orthogonal projection onto ∑_{1}^{M(h)} A^{(j)}(h)H, π′(h) = 1 − π(h). Then π′(h)χ(z − P (h))^{−1}χ is holomorphic for z in a neighborhood of Ω_{6}(h). On the other hand, we have

∥π′(h)χ(z − P)^{−1}χ∥_{H→H} ≤ Ae^{Ah^{−n^{♯}} log \frac{1}{hS(h)}}  (11.8)

in Ω_{6}(h) \ ∪_{j=1}^{M(h)}D(z_{j}, hS(h)). Put

Ω_{5}(h) = [a(h) − 5h^{k}, b(h) + 5h^{k}] + i[−S(h)h^{−1−n^{♯}}, S(h)].  (11.9)

Then (11.8), the maximum principle and the upper bound on M(h) imply that (11.8) holds in all of Ω_{5}(h), without any discs removed. (Notice that we cannot have any connected chain of discs D(z_{j}, hS(h)) from Ω_{5}(h) to ∂Ω_{6}(h) \ R.)

Now for Im z > 0, we have

∥π′(h)χ(z − P (h))^{−1}χ∥_{H→H} ≤ \frac{1}{|Im z|},  (11.10)

and together with (11.8) and Lemma 11.4, this implies:

∥π′(h)χ(z − P (h))^{−1}χ∥_{H→H} ≤ \frac{B}{S(h)}, z ∈ [a(h) − h^{k}, b(h) + h^{k}]. (11.11)

Now, use the quasimodes and conclude that for Im z > 0:

π′(h)u_{j}(h) = π′(h)χ(z − P)^{−1}χ(z − P)u_{j},

where we also used that χ = 1 near supp u_{j}. Let z tend to E_{j} and obtain:

∥π′u_{j}∥ ≤ \frac{B}{S(h)}R(h) ≤ Bh^{n^{♯}+1}.

In other words,

πu_{j} = u_{j} + w_{j},

with

∥w_{j}∥ ≤ Bh^{n^{♯}+1}.

It follows that

a_{j,k} := (πu_{j}|πu_{k}) − δ_{j,k} = O(h^{n^{♯}+1}),

so the matrix 1 + (a_{j,k}) is invertible and hence (πu_{j})_{j=1}^{m(h)} are linearly independent. Hence dim R(π) ≥ m(h), so M(h) ≥ m(h). #

12 Microlocal analysis and dynamical bounds

In this chapter, we develop some amount of analytic microlocal analysis in order to study upper bounds for the number of resonances for second order operators in the semi-classical limit. In the case the coefficients are analytic everywhere, the first and most basic result (Theorem 12.13) is that if there are no trapped classical trajectories at some positive energy, say 1, then there is a complex neighborhood of 1, independent of h, which contains no resonances when h is small enough. The second main result (Theorem 12.15) is un upper bound on the number of resonances in certain possibly h-dependent neighborhoods of 1 in terms of a certain escape function, which measures the amount of trapped trajectories. When the classical dynamics is hyperbolic this leads to a more explicit estimate in terms of the dimension of the set of trapped trajectories. (See Theorem 12.22.). The first result is implicit in Helffer-Sjöstrand [38] and is really a direct consequence of the theory developed there which is a direct microlocal treatment of resonances by means of certain FBI-transforms and suitable weighted spaces. The other two results were obtained in [72]. The theory of [38] permits to treat operators with a quite general behaviour near infinity and that was of importance in some of the main applications there. In [72] we used the same theory as the general frame-work.

Lahmar-Benbernou and Martinez [10] noticed that if we retrict the attention to operators that converge to the Laplacian near infinity, then one can use a Bargman transform (which is a special case of FBI-transforms) after a simple complex scaling, and get a simpler version of the theory. This simplifies the treatment near infinity. We follow that approach here, which permits to get faster to the main ideas of [38], [72], and especially the approach of [72] is followed without any essential changes.

12.1 Trajectories and escape functions

We will consider a simplified version of scattering for classical particles (Simon, Hunziker) and the relation with the existence of certain escape functions. We follow the appendix of [29]. Consider the classical Hamiltonian

p(x, ξ) = ∑_{|α|≤2} a_{α}(x)ξ^{α},  (12.1)

where a_{α} ∈ C^{∞}(R^{n}; R), ∂_{x}^{β}a_{α}(x) = o_{β}(1)⟨x⟩^{−|β|}, x → ∞, β ≠ 0,

p(x, ξ) − ξ^{2} → 0, x → ∞,  (12.2)
∑_{|α|=2} a_{α}(x)ξ^{α} ≥ \frac{1}{C} |ξ|^{2}.  (12.3)

We shall mainly work near the energy surface p^{−1}(1).

Let

G(x, ξ) = x · ξ.  (12.4)

If p_{0}(x, ξ) = ξ^{2}, then H_{p_{0}}G = 2ξ · ∂_{x}(x · ξ) = 2ξ^{2}, which is ≥ 1/C on p_{0}^{−1}([1−ϵ, 1+ϵ]) if ϵ ≤ 1/2. If we restrict the attention to p^{−1}([1−ϵ, 1+ϵ]) =: Σ_{p}^{ϵ}, then

|(p′_{ξ} − p′_{0,ξ})(x, ξ)| = o(1), |(p′_{x} − p′_{0,x})(x, ξ)| = o(1)⟨x⟩^{−1}, |x| → ∞,

while G′_{x}, ⟨x⟩^{−1}G′_{ξ} remain bounded, so

|(H_{p}G − H_{p_{0}}G)(x, ξ)| → 0, Σ_{p}^{ϵ} ∋ (x, ξ) → ∞.  (12.5)

Also note that the H_{p}-flow is complete in the sense that a trajectory t ↦ exp(tH_{p})(x, ξ) cannot reach infinity in finite time.

Fix ϵ ∈]0, \frac{1}{2}]. Clearly there is a compact set K̃ ⊂ Σ_{p}^{ϵ}, such that

H_{p}G ≥ \frac{1}{C̃} , on Σ_{p}^{ϵ} \ K̃,  (12.6)

for some C̃ > 0. Choose T > 0 large enough so that

K̃ ⊂ {ρ ∈ Σ_{p}^{ϵ}; 1 − T < G(ρ) < T − 1}.  (12.7)

Define the outgoing (+) and incoming (–) tails:

Γ_{+} = {ρ ∈ Σ_{p}^{ϵ}; exp(tH_{p}) ↛ ∞, t → −∞},
Γ_{−} = {ρ ∈ Σ_{p}^{ϵ}; exp(tH_{p}) ↛ ∞, t → +∞}.

Notice that if ρ ∈ Σ_{p}^{ϵ} \ Γ_{+}, then G(exp tH_{p}(ρ)) ≤ 1 − T for some t(ρ) < 0, and that G(exp tH_{p}(ρ)) ↘ −∞, t(ρ) > t → −∞. It follows that Γ_{+} is closed (and the same holds for Γ_{−}).

Proposition 12.1 For every s ∈ R, Γ_{+} ∩{ρ ∈ Σ_{p}^{ϵ}; G(ρ) ≤ s} and Γ_{−} ∩{ρ ∈ Σ_{p}^{ϵ}; G(ρ) ≥ s} are compact.

Proof. We only consider the first of the two sets. Notice that Γ_{+} ∩ {ρ ∈ Σ_{p}^{ϵ}; G(ρ) ≤ s} is empty when s ≤ 1 − T. Clearly there is a compact set K̂ ⊂ Σ_{p}^{ϵ}, depending on s, such that if ρ ∈ K̃ and |t| ≤ (s + T − 1)C̃, then exp tH_{p}(ρ) ∈ K̂. Then if μ ∈ Σ_{p}^{ϵ} \ K̂, G(μ) ≤ s, we know that exp(tH_{p})(μ) reaches the region G ≤ 1 − T for some t ∈ [−C̃(s + T − 1), 0]. Hence μ ∉ Γ_{+}.

#

Define the trapped set K := Γ_{+} ∩Γ_{−}. It is easy to see that K is compact.

Proposition 12.2 If Γ_{−} ≠ ∅ (or if Γ_{+} ≠ ∅), then K ≠ ∅.

Proof. Let ρ ∈ Γ_{−}. Then {exp tH_{p}(ρ); t ≥ 0} is contained in a compact set L, so we have exp t_{j}H_{p}(ρ) → ρ_{0} ∈ Σ_{p}^{ϵ}, for some sequence t_{j} → +∞. Put ρ_{j} = exp t_{j}H_{p}(ρ). For every S > 0, we have exp tH_{p}(ρ_{j}) → exp tH_{p}(ρ_{0}) uniformly for |t| < S. But exp tH_{p}(ρ_{j}) = exp(t_{j} + t)H_{p}(ρ_{j}), for |t| < S and j large enough so that t_{j} ≥ S. Hence exp tH_{p}(ρ_{0}) ∈ L, |t| < S. Since S can be chosen arbitrarily large, we conclude that ρ_{0} ∈ K. #

Though we will not need it in the following, we also consider the true tails:

T_{±} = Γ_{±} \ K.

On Σ_{p}^{ϵ} we have the symplectic volume which is H_{p}-invariant. For any s < T − 1, the set T_{−} ∩ {ρ ∈ Σ_{p}^{ϵ}; G ≥ s} is bounded and hence of finite volume. On the other hand, for s < 1 − T, we have

exp(tH_{p})(T_{−} ∩ {G ≥ s}) ↘ ∅, t → +∞.

Hence

Vol (T_{−} ∩ {G ≥ s}) = Vol (exp(tH_{p})(T_{−} ∩ {G ≥ s})) → 0, t → +∞.

We get:

Proposition 12.3

Vol (T_{±}) = 0.

In [29] we also proved:

Proposition 12.4 The following three statements are equivalent:

(a) T_{+} ≠ ∅

(b) T_{−} ≠ ∅

(c) If K_{α} = {ρ ∈ Σ_{p}^{ϵ}; dist (ρ, K) ≤ α}, then K and K_{α} \ K are non-empty for every α > 0.

Example. A potential well in an island. [[[Elaborate.]]]

We next construct a modification G̃ of G with G̃ = G outside a bounded set and with H_{p}G̃ ≥ 0 everywhere on Σ_{p}^{ϵ}. Increasing K̃ if necessary, we may assume that K ⊂ K̃. Let H_{T} = {ρ ∈ Σ_{p}^{ϵ}; G(ρ) = T }. We have a diffeomorphism κ_{+} : R × H_{T} → Σ_{p}^{ϵ} \ Γ_{−}, given by κ_{+}(t, ρ) = exp(tH_{p})(ρ). Similarly, we have a diffeomorphism κ_{−} : R × H_{−T} → Σ_{p}^{ϵ} \ Γ_{+} (with the obvious definition of H_{−T}).

Choose 0 < f_{+} ∈ C_{0}^{∞}(Σ_{p}^{ϵ} \ Γ_{−}) such that f_{+} ≥ H_{p}(G) with equality in {G ≥ T } and outside a compact set in {−T ≤ G ≤ T }. Let G_{+} ∈ C^{∞}(Σ_{p}^{ϵ} \ Γ_{−}) be the solution of

H_{p}G_{+} = f_{+}, G_{+}_{|}_{H_{T}} = T.

Then G_{+} = G in {G ≥ T } and outside a compact set in {−T ≤ G ≤ T }. We also have G_{+} ≤ G and choosing f_{+} large enough, we may assume that

lim sup_{ρ→Γ_{−}∪H_{−T}} G_{+}(ρ) ≤ −T.  (12.8)

Let f_{−}, G_{−} have the analogous properties.

Choose θ_{±} ∈ C^{∞}(R; [0, 1]) with 1 = θ_{+} + θ_{−}, supp θ_{+} ⊂]1 − T, +∞[, supp θ_{−} ⊂] − ∞, T − 1[, and put

χ_{+}(t) = ∫_{−∞}^{t} θ_{+}(s)ds, χ_{−}(t) = ∫_{+∞}^{t} θ_{−}(s)ds.

We may further arrange so that θ_{−}(−t) = θ_{+}(t). Then χ′_{±} = θ_{±} and we have

supp χ_{+} ⊂]1 − T, +∞[, supp χ_{−} ⊂] − ∞, T − 1[,
χ_{+}(t) + χ_{−}(t) = t.

Put G̃ = χ_{+} ∘ G_{+} + χ_{−} ∘ G_{−} and notice that χ_{+} ∘ G_{+} extends to a smooth function (= 0) near {G ≤ −T } ∪ Γ_{−}. Similarly χ_{−} ∘ G_{−} is smooth and welldefined on all of Σ_{p}^{ϵ}. By construction we have G̃ = G in {G ≤ −T } ∪ {G ≥ T } ∪ ({−T < G < T } \ a compact set).

We have

H_{p}G̃ = (χ′_{+} ∘ G_{+})f_{+} + (χ′_{−} ∘ G_{−})f_{−} ≥ 0,  (12.9)
∇G̃ = (χ′_{+} ∘ G_{+})∇G_{+} + (χ′_{−} ∘ G_{−})∇G_{−},  (12.10)

and get

Proposition 12.5 Given G = x · ξ, we can find a new “escape” function G̃ ∈ C^{∞}(R^{2n}; R), equal to G outside a compact set, such that on Σ_{p}^{ϵ} we have G̃ = 0 in a neighbohood of K = Γ_{+} ∩ Γ_{−} and such that locally uniformly on Σ_{p}^{ϵ}:

H_{p}(G̃) ≥ \frac{1}{C_{0}} |∇G̃|.  (12.11)

We may arrange so that H_{p}(G̃) > 0 in Σ_{p}^{ϵ} outside an arbitrarily small neighborhood of K, and so that H_{p}(G̃) > 0 everywhere on Σ_{p}^{ϵ} in the case when K = ∅.

12.2 Bargmann transforms and pseudodifferential operators

We first review some well-known facts about Bargmann transforms (which can be viewed as linearized models for FBI-tranforms) following [78]. Let ϕ(x, y) be a holomorphic quadratic form (i.e. a homogenous polynomial of degree 2) on C_{x}^{n} × C_{y}^{n} such that

det ϕ′′_{xy} ≠ 0, Im ϕ′′_{yy} > 0.  (12.12)

In the case of a classical Bargmann transform, we have ϕ(x, y) = \frac{i}{2}(x − y)^{2}. Put

Φ(x) = sup_{y∈R^{n}} −Im ϕ(x, y)  (12.13)

Notice that the supremum is attained at a unique point y(x) ∈ R^{n}. In the special case above, we get Φ(x) = \frac{1}{2}(Im x)^{2}.

For u ∈ S′(R^{n}), 0 < h ≤ 1, the function

T u(x; h) = Ch^{−3n/4} ∫ e^{iϕ(x,y)/h}u(y)dy, x ∈ C^{n}  (12.14)

is entire and satisfies |T u(x; h)| ≤ O_{h}(1)⟨x⟩^{N_{u}}e^{Φ(x)/h} for some N_{u} ∈ R. If u ∈ S(R^{n}), then |T u(x; h)| ≤ O_{N,h}(1)⟨x⟩^{−N}e^{Φ(x)/h} for every N ∈ R.

We view T as a semiclassical Fourier integral operator with associated linear canonical transformation

κ_{T} : C^{2n} ∋ (y, −ϕ′_{y}(x, y)) ↦ (x, ϕ′_{x}(x, y)) ∈ C^{2n}.  (12.15)

For every x ∈ C^{n} there is a unique point (y(x), η(x)) ∈ R^{2n} with π_{x} ∘ κ_{T} (y(x), η(x)) = x, where π_{x} is the natural projection C_{x,ξ}^{2n} → C_{x}^{n}. Indeed, y(x) must be the same as above and η(x) = −ϕ′_{y}(x, y(x)). Let ξ(x) ∈ C^{n} be the point with (x, ξ(x)) = κ_{T} (y(x), η(x)), so that ξ(x) = ϕ′_{x}(x, y(x)).

Compare with

\frac{2}{i} \frac{∂Φ}{∂x}(x) = \frac{2}{i} \frac{∂}{∂x}(−Im ϕ(x, y(x))) = \frac{2}{i} \frac{∂}{∂x}(−Im ϕ)_{y=y(x)} =

\frac{2}{i} \frac{∂}{∂x}(\frac{i}{2}(ϕ(x, y) − \overline{ϕ(x, y)}))_{y=y(x)} = (\frac{∂}{∂x}ϕ)(x, y(x)) = ξ(x).

We conclude that

κ_{T} (R^{2n}) = Λ_{Φ} := \{(x, \frac{2}{i} \frac{∂Φ}{∂x}(x)); x ∈ C_{x}^{n}\}.  (12.16)

Since κ_{T} is canonical for the complex symplectic form σ = ∑_{1}^{n} dη_{j} ∧ dy_{j}, and R^{2n} is I-Lagrangian, i.e. a Lagrangian manifold for the real symplectic form Im σ, the same holds for σ_{|}_{Λ_{Φ}}. Further, the real 2-form σ_{|}_{R^{2n}} is nondegenerate, so the same holds for σ_{|}_{Λ_{Φ}}. We say that Λ_{Φ}, like R^{2n} is an IR-manifold: Lagrangian for Im σ and symplectic for the restriction of Re σ.

What does this means in terms of Φ? In general, let F (x) be a smooth real-valued function on some open set in C^{n} and define Λ_{F} as above. Consider the restriction to Λ_{F} of the fundamental one form ξ · dx (whose exterior differential equals σ):

ξ · dx_{|}_{Λ_{F}} ≃ \frac{2}{i} \frac{∂F}{∂x} · dx ≃ \frac{2}{i}∂F,  (12.17)

−Im ξ · dx_{|}_{Λ_{F}} ≃ \frac{i}{2}(\frac{2}{i} \frac{∂F}{∂x} · dx − \frac{2}{−i} \frac{∂F}{∂\overline{x}} · d\overline{x}) = dF,

so

−Im σ_{|}_{Λ_{F}} = d(−Im ξ · dx)_{|}_{Λ_{F}} ≃ d^{2}F = 0.

Hence Λ_{F} is I-Lagrangian for all sufficiently smooth F. Conversely, every smooth I-Lagrangian manifold of the form ξ = ξ(x) is locally of the form Λ_{F} for a suitable smooth function F (exercise). From (12.17) we get

σ_{|}_{Λ_{F}} ≃ d(\frac{2}{i} \frac{∂F}{∂x} · dx) = \frac{2}{i} ∑_{j,k} \frac{∂^{2}F}{∂\overline{x}_{j}∂x_{k}} d\overline{x}_{j} ∧ dx_{k} (= \frac{2}{i}\overline{∂}∂F).  (12.18)

We check that Λ_{F} is R-symplectic iff det(\frac{∂^{2}F}{∂\overline{x}∂x}) ≠ 0. Returning to Λ_{Φ} we then know that det \frac{∂^{2}Φ}{∂\overline{x}∂x} ≠ 0. On the other hand, (12.13) tells us that Φ(x) is the supremum of a family of pluriharmonic functions, so it is plurisubharmonic and then also strictly plurisubharmonic:

\frac{∂^{2}Φ}{∂\overline{x}∂x} > 0.  (12.19)

Remark. IR-manifolds in C_{x,ξ}^{2n} are totally real of maximal real dimension 2n. (More details should be given here in later versions, and in the meanwhile we refer to [73, 78].)

We shall show that T is bounded: L^{2}(R^{2}) → H_{Φ}(C^{n}) := Hol (C^{n}) ∩ L^{2}(C^{n}; e^{−2Φ(x)/h}L(dx)), where L(dx) denotes the Lebesgue measure on C^{n} and Hol (Ω) denotes the space of holomorphic functions on the open set Ω ⊂ C^{n}. The formal adjoint of T is given by

T^{∗}v(y) = \overline{C}h^{− \frac{3n}{4}} ∫ e^{−iϕ^{∗}(\overline{x},y)/h}v(x)e^{−2Φ(x)/h}L(dx),  (12.20)

where ϕ^{∗}(x, y) := \overline{ϕ(\overline{x}, \overline{y})} is holomorphic. Then we get for v ∈ Hol (C^{n}), with |v| ≤ O_{N,h}(1)⟨x⟩^{−N}e^{Φ(x)/h} for all N:

T T^{∗}v(x) = |C|^{2}C_{ϕ}h^{− \frac{3n}{2} + \frac{n}{2}} ∫ e^{\frac{2}{h} Ψ(x,\overline{z})}u(z)e^{− \frac{2}{h} Φ(z)}L(dz), C_{ϕ} > 0, (12.21)

where

Ψ(x, w) = vc._{y} \frac{i}{2}(ϕ(x, y) − ϕ^{∗}(w, y)).

Here “vc._{y}” means “critical value with respect to y of” .

Ψ(x, w) is holomorphic in both variables and

Ψ(x, \overline{x}) = vc._{y} \frac{i}{2}(ϕ(x, y) − ϕ^{∗}(\overline{x}, y)) = Φ(x).  (12.22)

Consider the strictly plurisubharmonic quadratic form

F (x, y) = \frac{1}{2}(Φ(x)+Φ(\overline{y}))−Re Ψ(x, y) = \frac{1}{2}Φ(x)+\frac{1}{2}Φ(\overline{y})−\frac{1}{2}Ψ(x, y)−\frac{1}{2}\overline{Ψ(x, y)},

which vanishes on the anti-diagonal D = \{y = \overline{x}\}. Computing ∂_{x} and ∂\overline{x} of (12.22), we get

∂_{x}Ψ = ∂_{x}Φ, ∂_{y}Ψ = ∂\overline{x}Φ on D.

It follows that ∂_{x}F = 0, ∂\overline{y}F = 0 on D and since F is real-valued, that ∇F = 0 on D. The strict plurisubharmonicity of F now implies that F (x, y) ∼ |x − \overline{y}|^{2}, or in other words,

−Φ(x) + 2Re Ψ(x, \overline{z}) − Φ(z) ∼ −|x − z|^{2}.  (12.23)

Let Πu denote the right hand side of (12.21). Using (12.23) we see that

Π = O(1) : L^{2}(e^{−2Φ/h}L(dx)) → Hol ∩ L^{2}(e^{−2Φ/h}L(dx)) =: H_{Φ}.

Taking ∂_{x}∂\overline{x} of (12.22) and using (12.19), we get

det ∂_{x}∂_{y}Ψ(x, y) ≠ 0.

For λ > 0, u ∈ Hol (C^{n}), |u| ≤ O_{h}(1)⟨x⟩^{N_{0}}e^{Φ(x)/h}, consider

I_{λ}u(x) = \frac{1}{(2πh)^{n}} ∫∫_{Γ_{λ}(x)} e^{\frac{i}{h} (x−y)·θ}u(y)dydθ,  (12.24)

Γ_{λ}(x) : θ = \frac{2}{i} \frac{∂Φ}{∂x}(\frac{x + y}{2} ) + i\frac{λ}{2}\overline{(x − y)}.

It is easy to see that the integral converges and is independent of λ (by Stokes’ formula). Letting first λ become very large we can further replace Γ_{λ} by

Γ_{t,λ} : θ = t\frac{2}{i} \frac{∂Φ}{∂x}(\frac{x + y}{2} ) + i\frac{λ}{2}\overline{(x − y)}, 0 ≤ t ≤ 1.

Then let λ → +∞ and consider

I_{0,λ}u(x) = \frac{1}{(2πh)^{n}} ∫∫_{Γ_{0,λ}(x)} e^{\frac{i}{h} (x−y)·θ}u(y)dydθ

= \frac{λ^{n}}{(2πh)^{n}} ∫∫ e^{− \frac{λ}{h} |x−y|^{2}}u(y)L(dy) → u(x), λ → ∞.

We obtain

I_{λ}u(x) = u(x), u ∈ H_{Φ}.  (12.25)

On the other hand, we can use the Kuranishi trick to write I_{λ}u(x) as

I_{λ}u(x) = \frac{C}{h^{n}} ∫∫_{_{Γ_{λ}}(x)} e^{\frac{2}{h} (Ψ(x,z)−Ψ(y,z))}u(y)dydz.  (12.26)

This is done by a change of variables, using that 2(Ψ(x, z) − Ψ(y, z)) = i(x − y) · θ̃(x, y, z) with θ̃(x, y, z) = \frac{2}{i}Ψ′_{x}(\frac{x+y}{2}, z). Here Γ̃_{λ}(x) is the image contour in the (y, z)-variables.

If we replace Γ̃_{λ}(x) by the contour z = \overline{y} (which can be justified by means of Stokes’ formula), we get

\frac{C}{h^{n}} ∫∫ e^{\frac{2}{h} Ψ(x,\overline{y})}u(y)e^{− \frac{2}{h} Φ(y)}dyd\overline{y},  (12.27)

which is a non-vanishing constant times Πu(x) in (12.21). Both Π and (12.27) are positive semi-definite on H_{Φ}, so C in (12.27) is C > 0. Adjusting the contant C in the definition of T, we get Πu(x) precisely. Using Stokes’ formula, we check quite easily that Πu(x) = I_{λ}u(x) for u ∈ Hol (C^{n}) with |u(x)| ≤ C(h)⟨x⟩^{N_{0}}e^{Φ(x)/h} for some N_{0} > 0. In view of (12.25) we have shown that

Πu = u,  (12.28)

for such functions u, and in particular for u ∈ H_{Φ}. It is also clear that Π is self-adjoint in L_{Φ}^{2} := L^{2}(C^{n}; e^{−2Φ/h}L(dx)). Finally Π maps the latter space into H_{Φ}, so

Π is the orthogonal projection : L_{Φ}^{2} → H_{Φ}.  (12.29)

We next claim that

T is unitary: L^{2}(R^{n}) → H_{Φ}.  (12.30)

We have already seen that T is bounded: L^{2}(R^{n}) → H_{Φ} and that T T^{∗} = 1 on H_{Φ}, so it only remains to see that T is injective: If T u = 0, then (∂_{x}^{α}T u)(0) = 0 for all α ∈ N^{n}, and hence

∫ e^{\frac{i}{h} ϕ(0,y)}p(y)u(y)dy = 0,

for every polynomial p. If we let F denote the Fourier transform, then this means that

p(D_{η})(F(u(y)e^{\frac{i}{h} ϕ(0,y)}))(0) = 0,

for all polynomials p, and hence that F(u(y)e^{\frac{i}{h} ϕ(0,y)})(η) = 0, since this function is entire.

We next look at the action of pseudodifferential operators. Let S(Λ_{Φ}, m) be the space of all C^{∞}-functions a on Λ_{Φ} such that ∂^{α}a = O_{α}(m) for every multiindex α, when identifying Λ_{Φ} linearly with R^{2n}. Here m is an order function with

m(x) ≤ C⟨x − y⟩^{N_{0}}m(y), x, y ∈ Λ_{Φ},

for some constants C, N_{0} > 0. When our symbols depend on h, it is assumed that the above estimates are uniform.

If u ∈ O_{N,h}(1)⟨x⟩^{−N}e^{Φ(x)/h} for every N, we put

Op_{h}(a)u(x) = \frac{1}{(2πh)^{n}} ∫∫_{Γ(x)} e^{\frac{i}{h} (x−y)·θ}a(\frac{x + y}{2} , θ)u(y)dydθ,  (12.31)

where Γ(x) is the contour: θ = \frac{2}{i} \frac{∂Φ}{∂x}(\frac{x+y}{2}). Notice that on this contour,

−Φ(x) + Re (i(x − y) · θ) + Φ(y) = 0,

so the integral converges. Using this parametrization, we get

Op_{h}(a)u(x) = h^{−n} ∫ k(x, y; h)u(y)dy,

where ∂\overline{x}k = ∂\overline{y}k, so by integration by parts, we see that Op_{h}(a)u(x) is holomorphic if u is.

Let m̃ be a second order function on Λ_{Φ}. Both m and m̃ will be viewed as functions on C_{x}^{n}, by the natural identification π_{x} : Λ_{Φ} → C_{x}^{n}, where π_{x} : (x, ξ) ↦ x is the projection onto the x-space. Let

L_{Φ,_{m}}^{2} = L^{2}(C^{n}; m̃^{2}e^{−2Φ/h}L(dx)), H_{Φ,_{m}} = Hol(C^{n}) ∩ L_{Φ,_{m}}^{2}.

Proposition 12.6 Op_{h}(a) extends to a bounded operator: H_{Φ,_{m}} → H_{Φ, \frac{_{m}}{m}}, whose norm is uniformly bounded with respect to h.

Proof. For 0 ≤ t ≤ 1, let Γ_{t}(x) be the contour

θ = \frac{2}{i} \frac{∂Φ}{∂x}(\frac{x + y}{2} ) + it\frac{\overline{x − y}}{⟨x − y⟩},

parametrized by y ∈ C^{n} and let G_{[0,1]}(x) be the (n + 1)-contour obtained by letting also t ∈ [0, 1] be a parametrizing variable.

Now we have an almost holomorphic extension ã of a to C^{∞}(C^{2n}) such that

\overline{∂}ã(x, ξ) = O_{N}(1)m(x)|ξ − \frac{2}{i} \frac{∂Φ}{∂x}|^{N}, |ξ − \frac{2}{i} \frac{∂Φ}{∂x}| ≤ O(1),

for every N ≥ 0 and such that ã(x, ξ) = O(1)m(x). For simplicity we shall write ã instead of a. Then Stokes’ formula gives

Op_{h}(a)u(x) = \frac{1}{(2πh)^{n}} ∫∫_{Γ_{1}(x)} e^{\frac{i}{h} (x−y)·θ}a(\frac{x + y}{2} , θ)u(y)dydθ (12.32)

+\frac{1}{(2πh)^{n}} ∫_{G_{[0,1]}(x)} e^{\frac{i}{h} (x−y)·θ}u(y)∂_{\overline{y},\overline{θ}}(a(\frac{x + y}{2} , θ)) ∧ dydθ.

For the first term to the right, we notice that along Γ_{1}(x) we have dydθ = O(1)dyd\overline{y} and

−Φ(x) + Re (\frac{i}{h}(x − y) · θ) + Φ(y) = −\frac{|x − y|^{2}}{⟨x − y⟩} .

Hence the first term to the right in (12.32) can be written

\frac{m(x)}{m̃(x)}e^{\frac{Φ(x)}{h}} h^{−n} ∫ e^{− \frac{1}{h} \frac{|x−y|^{2}}{⟨x−y⟩}} f(x, y; h)u(y)e^{− \frac{Φ(y)}{h}} m̃(y)L(dy),

with

f(x, y; h) = O(1)m(\frac{x + y}{2} )\frac{m̃(x)}{m(x)m̃(y)} ≤ O(1)⟨x − y⟩^{N_{0}},

so the reduced kernel

h^{−n}e^{− \frac{1}{h} \frac{|x−y|^{2}}{⟨x−y⟩}} f(x, y; h)

is of modulus

≤ Ch^{−n}e^{− \frac{1}{h} \frac{|x−y|^{2}}{⟨x−y⟩}} ⟨x − y⟩^{N_{0}}

which is the kernel of a uniformly bounded convolution operator: L^{2} → L^{2}. Thus the first term of the right hand side of (12.32) is

O(1) : L_{Φ,_{m}}^{2} → L_{Φ, \frac{_{m}}{m}}^{2} .

For the last term in (12.32), we notice that along G_{[0,1]}(x):

∂_{\overline{y},\overline{θ}}(a(\frac{x + y}{2} , θ)) ∧ dydθ = O_{N}(1)m(\frac{x + y}{2} )t^{N} \frac{|x − y|^{N}}{⟨x − y⟩^{N}} L(dy)dt,

for every N ≥ 0.

For every t we then get the reduced operator

v ↦ h^{−n} ∫ e^{− \frac{t}{h} \frac{|x−y|^{2}}{⟨x−y⟩}} k_{t}(x, y; h)v(y)L(dy),

with

|k_{t}(x, y; h)| = O_{N}(1)m(\frac{x + y}{2} )\frac{m̃(x)}{m(x)} \frac{1}{m̃(y)}t^{N} \frac{|x − y|^{N}}{⟨x − y⟩^{N}} ≤ O_{N}(1)t^{N} \frac{|x − y|^{N}}{⟨x − y⟩^{N}} |x−y|^{N_{0}},

for some fixed N_{0} > 0. We estimate the L^{2}-norm of this operator by the L^{1}-norm of the corresponding convolution kernel:

O_{N}(1)h^{−n} ∫ e^{− \frac{t}{h} \frac{|x|^{2}}{⟨x⟩}} t^{N} \frac{|x|^{N+N_{0}}}{⟨x⟩^{N}} L(dx).

Cut the integral into two parts: |x| ≤ 1, |x| ≥ 1: For ∫_{|x|≤1} ... we get the bound:

O(1)h^{−n} ∫_{0}^{∞} e^{− \frac{t}{2h} r^{2}}t^{N}r^{N+N_{0}+2n−1}dr =
O(1)h^{−n} ∫_{0}^{∞} e^{− \frac{tr^{2}}{2h}} (\frac{tr^{2}}{h})^{\frac{N+N_{0}+2n}{2}} \frac{dr}{r} · h^{\frac{N+N_{0}+2n}{2}} t^{N− \frac{N+N_{0}+2n}{2}} = O(1)h^{\frac{N+N_{0}}{2}} ,

if N is large enough.

For ∫_{|x|≥1}, we get the bound:

O(1)h^{−n} ∫_{1}^{∞} e^{− \frac{t}{2h} r}t^{N}r^{N_{0}+2n} \frac{dr}{r},

and the change of variables r = \frac{h}{t}ρ gives

O(1)h^{−n} ∫ ∞_{\frac{t}{h}} e^{− \frac{ρ}{2}} (\frac{h}{t}ρ)^{N_{0}+2n} \frac{dρ}{ρ}t^{N} ≤

O(1)h^{−n} ∫ ∞_{\frac{t}{h}} e^{− \frac{ρ}{2}} ρ^{N_{0}+2n} \frac{dρ}{ρ}h^{N_{0}+2n}t^{N−N_{0}−2n}.

The last integral is O((1 + \frac{t}{h})^{−M}) for every M ≥ 0. For t ≤ h^{1/2}, we gain as many powers of h as we like from t^{N−N_{0}−2n}. For t > h^{1/2} the integral is O((\frac{t}{h})^{−M}) ≤ O(h^{M/2}) for all M and we also gain as many powers of h as we like.

We conclude that the last term in (12.32) defines an operator which is

O(h^{∞}) : L_{Φ,_{m}}^{2} → L_{Φ, \frac{_{m}}{m}}^{2} .

This completes the proof of the proposition. #

Exercise: Show that T S(R^{n}) = ∩_{N≥0}H_{Φ,⟨x⟩^{N}}. Hint: We already know that T S ⊂ ∩_{N≥0}.... It then suffices to show that T^{∗}(∩_{N≥0}...) ⊂ S, since T T^{∗} = 1 on H_{Φ}. Exercise: Show that ∩_{N≥0}H_{Φ,⟨x⟩^{N}} is dense in H_{Φ}. Hint: Let χ ∈ C_{0}^{∞}(C^{n}) be equal to 1 near 0 and consider Π(χ(\frac{x}{R})u), R → ∞.

We end this section by discussing the link with h-pseudodifferential operators on R^{n}. If m is an order function on R^{2n} and a ∈ S(R^{2n}, m), then we can define Op_{h}(a) : S(R^{n}) → S(R^{n}) and we have boundedness results in Sobolev spaces analogous to the H_{Φ,_{m}} above. A classical way of developing the theory for the Weyl quantization on R^{n} (see [21] for more details) is to observe that if ℓ(x, ξ) is a real linear form on R^{2n}, then Op_{h}(ℓ) = ℓ(x, hD_{x}) and that this operator is essentially self-adjoint from C_{0}^{∞}(R^{n}). Moreover

e^{−iℓ(x,hD_{x})} = Op_{h}(e^{−iℓ(x,ξ)}).  (12.33)

Imitating a proof of an important invariance property for the Weylquantization under conjugation by certain metaplectic operators, one then observes that if k is the linear form on Λ_{Φ} with

k ∘ κ_{T} = ℓ,  (12.34)

then

k(x, hD_{x})T u = T ℓ(x, hD_{x})u, ∀u ∈ S.  (12.35)

Now on the transform side, we have from (12.35) and the unitarity of T, that k(x, hD_{x}) is essentially self-adjoint on H_{Φ} from T S(R^{n}) and we also check that

k(x, D_{x}) = Op_{h}(k), e^{−ik(x,hD_{x})} = Op_{h}(e^{−ik(x,ξ)}).

Again from (12.35), it follows that

e^{−ik(x,hD_{x})}T = T e^{−iℓ(x,hD_{x})}.  (12.36)

If a ∈ S(R^{2n}) and b ∈ S(Λ_{Φ}) are related by b ∘ κ_{T} = a, then we can represent a as a superposition of linear exponentals of the type e^{−iℓ(x,ξ)} (by Fourier’s inversion formula) and we get a corresponding superposition of b in exponentials e^{−ik(x,ξ)}. Passing the quantizations, we get

Op_{h}(b) ∘ T = T ∘ Op_{h}(a).  (12.37)

By a density argument this relation extends to the case b ∈ S(Λ_{Φ}, m), a ∈ S(R^{2n}, m ∘ κ_{T}).

For the later applications, we will need very little calculus of pseudodifferential operators. For short presentations, see [32] for the classical way (with further references given there) with a maximal use of the method of stationary phase, and [21] for the approach sketched here with essentially no explicit use of stationary phase.

12.3 Pseudodifferential operators with holomorphic symbols

Assume that a(x, ξ) is holomorphic in a tubular neighborhood Λ_{Φ} +W of Λ_{Φ}, where W is an open bounded neighborhood of 0 ∈ C^{2n}, and of class S(m) there, where m is an order function defined first on Λ_{Φ} and then extended to a full neighborhood by putting m(x, ξ) = m(x, \frac{2}{i} \frac{∂Φ}{∂x}(x)).

In this case, we have no remainder term in a formula like (12.32) and we get

Op_{h}(a)u(x) = \frac{1}{(2πh)^{n}} ∫∫_{Γ_{c}(x)} e^{\frac{i}{h} (x−y)·θ}a(\frac{x + y}{2} , θ)u(y)dydθ, u ∈ H_{Φ,_{m}},

(12.38)

if c > 0 is small enough, so that Γ_{c}(x) ⊂ Λ_{Φ} + W.

Let

Φ̃(x) = Φ(x) + f(x),  (12.39)

where f ∈ C^{1,1}(C^{n}) has support in some fixed compact set and sufficiently small norm: ∥f∥_{C^{1,1}} = ∥∇f∥_{Lip}+sup_{C^{n}} |f(x)|, where ∥g∥_{Lip} = sup_{x≠y} |g(x)− g(y)|/|x − y|. Recall that the gradient of a Lipschitz function g belongs to L^{∞} and that ∥g∥_{Lip} = ∥∇g∥_{L^{∞}} (with the right choice of pointwise norm), so the same holds for the derivatives of functions of class C^{1,1}.

We can define the Hilbert space H_{_{Φ},_{m}}, which coincides with H_{Φ,_{m}} as a space. The corresponding norms are equivalent for every fixed h but not uniformly when h → 0. We have an associated Lipschitz manifold Λ_{Φ}: ξ = \frac{2}{i} \frac{δ_{Φ}}{∂x}(x) which coincides with Λ_{Φ} outside a compact set and which is close to Λ_{Φ} in the sense of Lipschitz graphs (since f is small in C^{1,1}).

If c > 0 is small enough, we can use Stokes’ formula to replace Γ_{c}(x) in (12.40) by

Γ̃_{c}(x) : θ = \frac{2}{i} \frac{∂Φ̃}{∂x}(\frac{x + y}{2} ) + ic\frac{\overline{x − y}}{⟨x − y⟩}.  (12.40)

Lemma 12.7 Op_{h}(a) is bounded: H_{_{Φ},_{m}} → H_{_{Φ},_{m}/m}, uniformly with respect to h.

Proof. On the level of the amplitudes, the estimates are the same as before, so we only have to check that along Γ̃_{c}(x), we have

|e^{\frac{1}{h} (i(x−y)·θ−_{Φ}(x)+_{Φ}(y))}| ≤ e^{− \frac{c}{2} \frac{|x−y|^{2}}{⟨x−y⟩}}  (12.41)

Along Γ̃_{c}(x), we have using notation from R^{2n} ≃ C^{n}:

Re (i(x − y) · θ) = ⟨∇Φ̃(\frac{x + y}{2} ), x − y⟩ − c\frac{|x − y|^{2}}{⟨x − y⟩} ,  (12.42)

while Taylor’s formula with integral remainder gives:

Φ̃(x) − Φ̃(y) − ⟨∇Φ̃(\frac{x + y}{2} ), x − y⟩ =  (12.43)

⟨∫_{0}^{1}(1 − t)(Φ̃′′(\frac{x + y}{2} + t\frac{x − y}{2} ) − Φ̃′′(\frac{x + y}{2} − t\frac{x − y}{2} ))dt \frac{x − y}{2} , \frac{x − y}{2} ⟩

= ⟨∫_{0}^{1}(1 − t)(f′′(\frac{x + y}{2} + t\frac{x − y}{2} ) − f′′(\frac{x + y}{2} − t\frac{x − y}{2} ))dt \frac{x − y}{2} , \frac{x − y}{2} ⟩.

Since the support of f belongs to some fixed compact set, we see that

∫ (1−t)(f′′(\frac{x + y}{2} +t\frac{x − y}{2} )−f′′(\frac{x + y}{2} −t\frac{x − y}{2} ))dt = O(1)∥f′′∥_{L^{∞}}/⟨x−y⟩.

Combining this with (12.42), (12.43), we get

Re (i(x − y) · θ) − Φ̃(x) + Φ̃(y) = −(c + O(1)∥f′′∥_{L^{∞}})\frac{|x − y|^{2}}{⟨x − y⟩} , (12.44)

which gives (12.41), when ∥f′′∥_{L^{∞}} is small enough. #

Although not necessary in the following, we end this section with a brief discussion of other quantizations. (See [78] for more details on the holomorphic side and [21] for the real case.) For t ∈ [0, 1], put (formally):

Op_{h,t}(a)u(x) = \frac{1}{(2πh)^{n}} ∫∫_{Γ_{t}(x)} e^{i(x−y)·θ/h}a(tx + (1 − t)y, θ)u(y)dydθ, (12.45)

where Γ_{t}(x): θ = \frac{2}{i} \frac{∂Φ}{∂x}(tx + (1 − t)y). Along Γ_{t} we get

−Φ(x) + Re (i(x − y) · θ) + Φ(y) = (2t − 1)Φ(x − y).

Hence,

Proposition 12.8 Let Φ be strictly convex and let a ∈ S(Λ_{Φ}, m). Then Op_{h,t}(a) is a well-defined and uniformly bounded operator: H_{Φ,_{m}} → H_{Φ,_{m}/m} for 0 ≤ t ≤ \frac{1}{2}. When 0 ≤ t < \frac{1}{2} it is enough to assume that a = O(m(x)) to have the same conclusion, and when t = 0 we even get a uniformly bounded operator: L_{Φ,_{m}}^{2} → H_{Φ,_{m}/m}.

Next we are interested in describing the same operator with different quantizations. This is very classical in the real case, when Op_{h,t} is welldefined for 0 ≤ t ≤ 1. If a_{t} ∈ S(R^{2n}, m) depends smoothly on t, we get for u ∈ S:

∂_{t}Op_{h,t}(a_{t})u(x)  (12.46)

= \frac{1}{(2πh)^{n}} ∫∫ e^{\frac{i}{h} (x−y)·θ}∂_{t}(a_{t}(tx + (1 − t)y, θ))u(y)dydθ

= \frac{1}{(2πh)^{n}} ∫∫ e^{\frac{i}{h} (x−y)·θ}(∂_{t}a_{t})(tx + (1 − t)y, θ)u(y)dydθ +

\frac{1}{(2πh)^{n}} ∫∫ e^{\frac{i}{h} (x−y)·θ}(x − y) · (∂_{x}a_{t})(tx + (1 − t)y, θ)u(y)dydθ,

where the last term equals

\frac{1}{(2πh)^{n}} ∫∫ (\frac{h}{i}∂_{θ})(e^{\frac{i}{h} (x−y)·θ}) · ∂_{x}a_{t}(tx + (1 − t)y, θ)u(y)dydθ = (12.47)

−\frac{1}{(2πh)^{n}} ∫∫ e^{\frac{i}{h} (x−y)·θ}(\frac{h}{i}∂_{θ} · ∂_{x}a_{t})(tx + (1 − t)y, θ)u(y)dydθ.

We conclude that Op_{h,t}(a_{t}) is independent of t if a_{t} fulfills the Schrödinger type equation

∂_{t}a_{t} = \frac{h}{i}(∂_{θ} · ∂_{x})a_{t}.  (12.48)

The Cauchy problem for this equation is well-posed in the sense that if a_{s} = a_{t=s} is given in S(R^{2n}, m) for some s ∈ [0, 1], then we get a unique solution a_{t} ∈ C^{∞}([0, 1]; S(R^{2n}, m)).

The same discussion can be carried out in the complex framework, except that (12.48) is now of heat equation type (on Λ_{Φ}) and can be solved only in the forward time direction. Thus if a_{s} ∈ S(Λ_{Φ}, m) is given, we can find a_{t} ∈ C^{∞}([s, 1]; S(Λ_{Φ}, m)) which is entire for t > s such that Op_{h,s}(a_{s}) = Op_{h,t}(a_{t}) for t ≥ s. Notice that Op_{h,0} maps weighted L^{2}-spaces into weighted L^{2}-spaces of holomorphic functions.

We point out that the strict convexity condition on Φ in the proposition dissappears if we choose to represent our operators as in (12.26). The case t = 0 gives Töplitz operators.

The above calculations show that if a ∈ S(Λ_{Φ}, m) then for s, t ∈ [0, 1] we have

Op_{h,t}(a) − Op_{h,s}(a) = O(h) : H_{Φ,_{m}} → H_{Φ,_{m}/m}.  (12.49)

If we also know that a has a holomorphic extension of class S(m) to a tubular neighborhood of Λ_{Φ}, then we can replace Φ by Φ̃ = Φ+f in the above result, provided that the C^{1,1}-norm of f is suffiently small.

12.4 Approximation by multiplication operators

The main result of this section is

Proposition 12.9 Let a be holomorphic and of class S(m) in a tubular neighborhood of Λ_{Φ}. Let Φ̃ ∈ C^{1,1} be a small perturbation of Φ as before. Then with ξ(x) = \frac{2}{i} \frac{∂_{Φ}}{∂x}(x):

Op_{h}(a)u(x) = a(x, ξ(x))u(x) + ∑_{1}^{n}(∂_{ξ_{j}}a(x, ξ(x)))(hD_{x_{j}} − ξ_{j}(x))u(x) + Ru(x),

(12.50)

where R is of norm O(h): H_{_{Φ},_{m}} → L_{_{Φ},_{m}/m}^{2}.

Proof. We write Op_{h}(a)u(x) in (12.38) with the contour (12.40). We also assume that m = m̃ = 1 for simplicty. By Taylor expansion, we get

a(\frac{x + y}{2} , θ) = a(x, ξ(x)) + (∂_{ξ}a)(x, ξ(x)) · (θ − ξ(x))  (12.51)

−\frac{1}{2}(∂_{x}a)(x, ξ(x)) · (x − y) + O(|x − y|^{2}) + |θ − ξ(x)|^{2}).

Here the third term to the right gives no contribution to (12.38):

∫∫ e^{\frac{i}{h} (x−y)·θ}(x − y) · (∂_{x}a)(x, ξ(x))u(y)dydθ =
(∂_{x}a(x, ξ(x)) · ∫∫ \frac{h}{i}∂_{θ}(e^{\frac{i}{h} (x−y)·θ}u(y))dydθ = 0.

The first two terms to the right in (12.51) lead to the corresponding terms in (12.50). Along the contour (12.40), we have θ − ξ(x) = O(|x − y|), so the remainder term in (12.51) gives contribution O(|x − y|^{2}) in (12.38) and that gives an operator R with norm O(h). #

We use the above result to study certain scalar products.

Proposition 12.10 Let a be as before and let ψ(x) be locally Lipschitz on C^{n} with ψ(x), |∇ψ(x)| = O(m_{ψ}(x)), where m_{ψ} together with m_{1}, m_{2} are order functions with m_{ψ}m ≤ m_{1}m_{2}. Then

(ψOp_{h}(a)u|v)_{L2 ; _{Φ}} = ∫ ψ(x)a(x, ξ(x))u(x)\overline{v(x)}e^{−2_{Φ}(x)/h}L(dx) (12.52)

+O(h)∥u∥_{H_{_{Φ},m_{1}}}∥v∥_{H_{_{Φ},m_{2}}}

for u ∈ H_{_{Φ},m_{1}}, v ∈ H_{_{Φ},m_{2}}.

That the action of a pseudodifferential operator can be approximated by a multiplication operator with an error O(h^{1/2}) after a suitable integral transform was used in [79] and in [19]. That the error becomes only O(h) for scalar products was observed in [19] and used there to give a short proof of the sharp Gårding inequality. Here we are in a more general situation, and the limited regularity in the weights is quite essential for the later applications. (We follow [74].)

Proof. Assume m_{ψ} = m = m_{1} = m_{2} = 1 for simplicity. Substituting (12.50) into (12.52), we see that we only have to estimate

∫ ψ(x)(∂_{ξ_{j}}a)(x, ξ(x))(hD_{x_{j}} − ξ_{j}(x))u(x)\overline{v(x)}e^{−2_{Φ}(x)/h}L(dx).  (12.53)

Here (∂_{ξ_{j}}a)(x, ξ(x)) is Lipshitz, so we can make an integration by parts in (12.53) and get O(h)∥u∥_{H_{Φ}}∥v∥_{H_{Φ}} plus

∫ ψ(x)(∂_{ξ_{j}}a)(x, ξ(x))u(x)\overline{v(x)}(−hD_{x_{j}} − ξ_{j}(x))(e^{−2_{Φ}(x)/h})L(dx), (12.54)

which vanishes, since (−hD_{x_{j}} − ξ_{j}(x))(e^{−2_{Φ}(x)/h}) = 0. We also used that hD_{x_{j}} \overline{v(x)} = 0 since v is holomorphic. #

This can be iterated. Let b be second symbol like a with associated order functions m_{b} and write m = m_{a} for the order function associated to a.

Proposition 12.11 Let a, b, ψ, m_{a}, m_{b}, m_{ψ}, m_{1}, m_{2} be as above with m_{ψ}m_{a}m_{b} ≤ m_{1}m_{2}. Then,

(ψOp_{h}(a)u|Op_{h}(b)v)_{L2 ; _{Φ}} =  (12.55)

∫ ψ(x)a(x, ξ(x))u(x)\overline{b(x, ξ(x))v(x)}e^{−2_{Φ}(x)/h}L(dx) + O(h)∥u∥_{H_{_{Φ},m_{1}}}∥v∥_{H_{_{Φ},m_{2}}},

for u ∈ H_{_{Φ},m_{1}}, v ∈ H_{_{Φ},m_{2}}.

Corollary 12.12 Under the appropriate part of the same assumptions, we have

∥Op_{h}(a)u∥_{H_{Φ}}^{2} = ∥a(x, ξ(x))u(x)∥_{L2 ; _{Φ}}^{2} + O(h)∥u∥_{H_{_{Φ},ma}}^{2},

for u ∈ H_{_{Φ},m_{a}}.

12.5 Absence of resonances in the analytic non-trapping case

Let

P = ∑_{|α|≤2} a_{α}(x; h)(hD_{x})^{α} : H^{2}(R^{n}) → L^{2}(R^{2})  (12.56)

satisfy the general assumptions for “Q” in Chapter 7, that we here recall: a_{α}(x; h) is independent of h for |α| = 2, a_{α}(·; h) is bounded in C_{b}^{∞}, uniformly with repect to h, and a_{α}(·; h) = a_{α,0}(·)+ O(h) in this space, where a_{α,0} is independent of h . ∑_{|α|=2} a_{α}(x)ξ^{α} ≥ \frac{1}{C} |ξ|^{2}. ∑_{|α|≤2} a_{α}(x; h)ξ^{α} → ξ^{2}, |x| → ∞, uniformly with respect to h.

In addition to the assumption that the coefficients extend holomorphically to some angle, we assume them to be analytic everywhere. Thus we assume

(A) There exists a constant C > 0 such that the coefficients a_{α} extend holomorphically in x to {x ∈ C^{n}; |Im x| < C^{−1}⟨Re x⟩}, and the relevant parts of the preceding assumptions extend to this complex domain.

Recall that if θ > 0 is small enough, then the resonances of P in e^{−i[0,2θ[}]0, +∞[ are defined to be the eigenvalues in this sector of P_{|}_{e^{iθ}R^{n}}.

Let p(x, ξ) = ∑_{|α|≤2} a_{α,0}(x)ξ^{α} and make the non-trapping assumption

(NT) There are no trapped H_{p}-trajectories in p^{−1}(1).

By a trapped trajectory we mean a (maximal) integral curve R ∋ t ↦ exp(tH_{p})(ρ_{0}) ∈ p^{−1}(1) such that | exp(tH_{p})(ρ_{0})| ↛ ∞ both when t → −∞ and when t → +∞. According to section 12.1, this means that “K” there is empty, if “ϵ” there is small enough. According to Proposition 12.5,, we can find an “escape” function

G(x, ξ) + f(x, ξ),  (12.57)

with f ∈ C_{0}^{∞}(R^{2n}; R), such that

H_{p}G > 0 on p^{−1}(1).  (12.58)

Consider the IR-manifold:

Λ_{ϵG} : Im (x, ξ) = ϵH_{G}(Re (x, ξ)), 0 < ϵ ≪ 1.  (12.59)

(We leave as an exercise to verify that this is really an IR-manifold.) Outside a bounded set, we have H_{G} = x·δ_{x} −ξ ·∂_{ξ} and here Λ_{ϵG} becomes:

Im x = ϵRe x, Im ξ = −ϵRe ξ.  (12.60)

Let Γ_{ϵ} ⊂ C^{n} be given by

Im x = ϵRe x,  (12.61)

so that Γ_{ϵ} = e^{iθ(ϵ)}R^{n}, with θ(ϵ) = arctg ϵ. Then the projection of Λ_{ϵG} \ (compact set) is equal to Γ_{ϵ} and we claim that Λ_{ϵG} coincides with T^{∗}Γ_{ϵ} outside a bounded set. (We view T^{∗}Γ_{ϵ} as a subset of C_{x,ξ}^{2n} as in Chapter 7.) Indeed for (x, ξ) ∈ Λ_{ϵG} \ (compact set), we have Im ξ = −ϵRe ξ in addition to (12.61) and hence

ξ · dx_{|}_{Γ_{ϵ}} = (1 − iϵ)Re ξ · (1 + iϵ)dRe x = (1 + ϵ^{2})Re ξ · dRe x

is real. Also notice that the parametrization κ : R^{n} ∋ y ↦ (1+iϵ)y = x ∈ Γ_{ϵ} induces an identification of T^{∗}R^{n} = R_{x,ξ}^{2n} and T^{∗}Γ_{ϵ} via the map

(y, η) ↦ (κ(y),^{t}dκ(y)^{−1}η) = ((1 + iϵ)y, (1 + iϵ)^{−1}η).

Next parametrize Λ_{ϵG} by R^{2n} ∋ (x, ξ) ↦ (x, ξ) + iϵH_{G}(x, ξ) and compute p_{ϵ} = p_{|}_{Λ_{ϵG}} by Taylor expansion:

p_{ϵ} ≃ p((x, ξ) + iϵH_{G}(x, ξ)) = p(x, ξ) − iϵH_{p}G(x, ξ) + O(ϵ^{2}⟨ξ⟩^{2}). (12.62)

This estimate is uniform since

H_{G} = O(⟨x⟩)\frac{∂}{∂x} + O(⟨ξ⟩)\frac{∂}{∂ξ} , ∂_{x}^{α}∂_{ξ}^{β}G = O(⟨x⟩^{1−|α|}⟨ξ⟩^{1−|β|})

∂_{x}^{α}∂_{ξ}^{β}p = O(⟨ξ⟩^{2−|β|}⟨x⟩^{−|α|}),

and we can work in boxes |x−x_{0}| < \frac{1}{2}⟨x_{0}⟩, |ξ −ξ_{0}| < \frac{1}{2}⟨ξ_{0}⟩, using the rescaled variables (y, η), given by

x = x_{0} + ⟨x_{0}⟩y, ξ = ξ_{0} + ⟨ξ_{0}⟩η,

for which

∂_{y}^{α}∂_{η}^{β}H_{G} = O(1)\frac{∂}{∂y} + O(1)\frac{∂}{∂η} , ∂_{y}^{α}∂_{η}^{β}p = O(⟨ξ_{0}⟩^{2}).

Using this together with (12.58), we see that for ϵ > 0 small enough, we have with the parametrization used in (12.62):

Re p_{ϵ}(x, ξ) = p(x, ξ)+O(ϵ⟨ξ⟩^{2}), Im p_{ϵ}(x, ξ) ≤ −\frac{ϵ}{C} +Cϵ|p(x, ξ)−1|. (12.63)

In particular, we have

|p_{ϵ}(x, ξ) − 1| ≥ ϵ⟨ξ⟩^{2}/C.  (12.64)

Let T : L^{2}(R^{n}) → H_{Φ_{0}^{0}}(C^{n}) be a Bargman transform as in (12.14). Replacing the integration contour R^{n} there by Γ_{ϵ}, we get a new Bargman tranform

T : L^{2}(Γ_{ϵ}) → H_{Φ_{ϵ}^{0}}(C^{n}),  (12.65)

by identifying Γ_{ϵ} with R^{n} in the natural way. Viewing Γ_{ϵ} ⊂ C^{n} and T^{∗}Γ_{ϵ} ⊂ C^{2n} as subspaces in the natural way, we justify the writing “T ” rather than “T_{ϵ}” and we have an associated canonical tranformation

κ_{T} : C^{2n} → C^{2n} with κ_{T} (T^{∗}Γ_{ϵ}) = Λ_{Φ_{ϵ}^{0}},  (12.66)

where Φ_{ϵ}^{0} = sup_{y∈Γ_{ϵ}} −Im ϕ(x, y). Clearly Φ_{ϵ}^{0}(x) = Φ_{0}^{0}(x) + O(ϵ)|x|^{2} is a quadratic form which is strictly plurisubharmonic.

Now recall that Λ_{ϵ} = Λ_{ϵG} coincides with T^{∗}Γ_{ϵ} outside a bounded set and that both manifolds are ϵ-perturbations of R^{2n} in the natural sense. Then κ_{T} (Λ_{ϵ}) is an IR-manifold, equal to Λ_{Φ_{ϵ}^{0}} outside a bounded set and ϵ-close to that manifold everywhere, so

κ_{T} (Λ_{ϵ}) = Λ_{Φ_{ϵ}},  (12.67)

where Φ_{ϵ}(x) = Φ_{ϵ}^{0}(x) + ϵg(x, ϵ), where g is C^{∞} in both arguments and has its x-support in a fixed compact set.

Theorem 12.13 Under the assumptions above, and in particular (A) and (NT), there exist constants h_{0}, C_{0} > 0 such that P has no resonances in the disc D(1, C_{0}^{−1}), for 0 < h < h_{0}.

Corollary 12.14 Let V ∈ C^{∞}(R^{n}; R) extend to a holomorphic function in |Im x| < ⟨Re ⟩/C which tends to 0 when |x| → ∞ in that domain. Then there exist C_{0}, C_{1} > 0, such that P = −Δ + V (x) has no resonances in {z ∈ C; Re z > C_{1}, Im z > −Re z/C_{0}}.

Proof of the Corollary. It will follow from the proof of the theorem, that we we have uniformity with respect to additional parameters if the assumptions hold uniformly.

Consider P − z with Re z ≫ 1, |Im z| < Re z/C_{0} and write this operator as

(Re z)(−h^{2}Δ + \frac{1}{Re z} V − 1 − \frac{iIm z}{Re z}), h^{2} = \frac{1}{Re z} .

Theorem 12.13 implies that we have no resonances when Re z is large and Im z/Re z is small. #

Proof of the theorem. Let ϵ > 0 be sufficiently small but fixed. Let

Q = T ∘ P_{|}_{Γ_{ϵ}} ∘ T^{−1}.  (12.68)

Then (using the same letters for symbols and operators)

Q = Op_{h}(Q) = Q^{w}(x, hD_{x}; h),  (12.69)

where on the symbol level,

(Q ∘ κ_{T})(x, ξ; h) = P (x, ξ; h).  (12.70)

Here P denote the Weyl-symbol of our operator P which is holomorphic in a tubular neighborhood of T^{∗}Γ_{ϵ} and satisfies P (x, ξ; h) = p(x, ξ) + O(h⟨ξ⟩^{2}) there. (See for instance [21].) We see that Q(·; h) is holomorphic in a tubular neighborhood of Λ_{Φ_{ϵ}^{0}} of the form Λ_{Φ_{ϵ}^{0}} + W, with W independent of ϵ, and in this neighborhood we have

Q(x, ξ; h) = O(m(x, ξ)), Q = q + O(hm),  (12.71)

where m is the order function with m∘κ_{T} = ⟨ξ⟩^{2}, and q is given by q∘κ_{T} = p. For ϵ > 0 small enough, Λ_{Φ_{ϵ}} is well inside Λ_{Φ_{ϵ}^{0}} + W, so we can consider

Q^{w} : H_{Φ_{ϵ},m} → H_{Φ_{ϵ}}.  (12.72)

Let z ∈ D(0, C_{0}^{−1}) with C_{0}^{−1} ≪ ϵ (ϵ being fixed). Then

|q(x, ξ) − z| ∼ m(x, ξ) on Λ_{Φ_{ϵ}},  (12.73)

and Corollary 12.12 together with (12.71) imply that

∥(Q^{w} − z)u∥_{H_{Φϵ}}^{2} ≥ \frac{(1 − O_{ϵ}(h))}{C_{ϵ}}∥u∥_{H_{Φϵ},m}^{2}.  (12.74)

If z is a resonance, let v ∈ H^{2}(Γ_{ϵ}) be a corresponding eigenfunction of P_{|}_{Γ_{ϵ}}, so that (P −z)v = 0 on Γ_{ϵ}. Applying T we get u := T v ∈ H_{Φ_{ϵ},m}, (Q^{w}−z)T v = 0 and (12.74) implies that if h > 0 is small enough, then u = T v = 0 and so v = 0 in contradiction with the assumption that z is a resonance. #

12.6 Dynamical upper bounds in terms of escape functions

We drop the non-trapping assumtion (NT), and keep all the other assumptions of the preceding section (including (A)). Let K be the union of trapped trajectories in Σ_{p}^{ϵ_{0}} = p^{−1}([1 − ϵ_{0}, 1 + ϵ_{0}]), where ϵ_{0} > 0 is small and fixed. Let G be a real-valued function on R^{2n}, such that

G and H_{p}G are of class C^{1,1}, G = x · ξ for |(x, ξ)| large, (12.75)
H_{p}G is ≥ O(1)^{−1}|∇G|^{2} locally uniformly in Σ_{p}^{ϵ_{0}},
H_{p}G > 0 outside a small neighborhood of K in Σ_{p}^{ϵ_{0}}.

In section 12.1 we saw that such smooth functions always exist, but our estimates will depend on a specific such function, and good choices of such functions seem to require limited regularity as we shall see in section 12.7.

For 0 ≤ δ ≤ r ≤ 1, δ ≤ C_{0}^{−1}, let M(r, δ) be the number of resonances of P in D(1 + ir, r + δ). Let W be a small neighborhood of K in Σ_{p}^{ϵ_{0}}, outside which H_{p}G is bounded from below by some positive constant. Put

V (r, δ) = Vol ({ρ ∈ W; (p(ρ) − 1)^{2} + 2H_{p}G + (H_{p}G)^{2} ≤ 2rδ + δ^{2}}) (12.76)
= Vol ({ρ ∈ W; |(p(ρ) − iH_{p}G) − (1 + ir)|^{2} ≤ (r + δ)^{2}}).

Notice that this quantity increases when r or δ increases. Also, if

Ṽ (r, δ) = Vol ({ρ ∈ W; (p(ρ) − 1)^{2} + H_{p}G ≤ rδ}),

then V (r, δ) ≤ Ṽ (r, 3δ), Ṽ (r, δ) ≤ V (r, 3δ).

Theorem 12.15 For C_{1}, C_{0} > 0 sufficiently large and for C_{0}h ≤ δ ≤ r ≤ 1, δ ≤ 1/C_{0}, rδ ≥ C_{0}h, we have

M(r, δ) ≤ CV (r, Cδ)h^{−n}.  (12.77)

Define Γ_{ϵ} as in (12.61) and let T : L^{2}(Γ_{ϵ}) → H_{Φ_{ϵ}^{0}}(C^{n}) be a Bargman transform as in section 12.5. As there, we define Φ_{ϵ}, so that

κ_{T} (Λ_{ϵG}) = Λ_{Φ_{ϵ}}, κ_{T} (T^{∗}Γ_{ϵ}) = Λ_{Φ_{ϵ}^{0}}, Φ_{ϵ} = Φ_{ϵ}^{0} + ϵg(·, ϵ),  (12.78)

where g(·, ϵ) is bounded in C^{1,1} when ϵ varies, and has uniformly compact support. (We recall the definition of Λ_{ϵG} below.)

Having already fixed a small ϵ_{0} > 0, we will let ϵ > 0 be small and (later) fixed. Let m be the order function with m ∘ κ_{T} = ⟨ξ⟩^{2}.

By abuse of notation, we let P also denote the conjugated operator T P_{|}_{Γ_{ϵ}}T^{−1}. Then (on the transform side)

P = O(1) : H_{Φ_{ϵ}^{0},m} → H_{Φ_{ϵ}^{0}}, H_{Φ_{ϵ},m} → H_{Φ_{ϵ}},

uniformly in ϵ, h.

Consider the Hermitian form:

q(u, u) = ((P − 1 − ir)u|(P − 1 − ir)u)_{H_{Φϵ}}, 0 < r ≤ 1,  (12.79)

formally equal to (Qu|u)_{H_{Φϵ}}, with Q = (P − 1 − ir)^{∗}(P − 1 − ir), where ∗ indicates that we take the adjoint in the Hilbert space H_{Φ_{ϵ}}.

Let Λ_{ϵG} be the the image of

R^{2n} ∋ (x, ξ) ↦ (x, ξ) + iϵH_{G}(x, ξ),  (12.80)

and identify R^{2n} and Λ_{ϵG} by means of this map. We also identify Λ_{ϵG} and Λ_{Φ_{ϵ}} by means of κ_{T}, and Λ_{Φ_{ϵ}} and C_{x}^{n} by means of the natural projection π_{x} : (x, ξ) ↦ x. Consequently we shall view p_{ϵ} := p_{|}_{Λ_{ϵ}} also as a function on R^{2n}, Λ_{Φ_{ϵ}} or C^{n} whenever convenient.

According to Corollary 12.12, we have

q(u, u) = ∫_{C^{n}} |p_{ϵ} − 1 − ir|^{2}|u|^{2}e^{−2Φ_{ϵ}/h}L(dx) + O(h)∥u∥_{H_{Φϵ},m}^{2}  (12.81)

(where p_{ϵ} is viewed as a function on C^{n}).

If we now consider p_{ϵ} as a function on R^{2n}, we know that the values of p_{ϵ} on the complement of Σ_{p}^{ϵ_{0}} and on the complement of W in Σ_{p}^{ϵ_{0}} avoid a set of the form {z; |Re z − 1| < C_{0}^{−1}, Im z > −ϵ/C_{0}} and consequently we have here

|p_{ϵ} − 1 − ir|^{2} ≥ (r + \frac{ϵ}{C_{0}} )^{2},  (12.82)

when ϵ > 0 is small enough. In the complement of Σ_{p}^{ϵ_{0}}, we even have

|p_{ϵ} − 1 − ir|^{2} ≥ r^{2} + \frac{⟨ξ⟩^{2}}{O(1)}.  (12.83)

In a neighborhood of K in Σ_{p}^{ϵ_{0}}, we take a closer look:

p_{ϵ} ≃ p((x, ξ)+iH_{G}(x, ξ)) = p(x, ξ)−iϵH_{p}G(x, ξ)+O(ϵ^{2})|∇G|^{2}+iO(ϵ^{3})|∇G|^{3},

(12.84)

where the first remainder term is real and the second one is purely imaginary. Hence, using (12.75):

−Im p_{ϵ} = ϵ(1 + O(ϵ^{2}))H_{p}G,  (12.85)
Re p_{ϵ} = p + O(ϵ^{2})H_{p}G = p + O(ϵ)Im p_{ϵ}.  (12.86)

It follows that near K in Σ_{p}^{ϵ_{0}}, we have

|p_{ϵ} − 1 − ir|^{2} = (Re p_{ϵ} − 1)^{2} + (r − Im p_{ϵ})^{2}

= (p − 1 + O(ϵ)Im p_{ϵ})^{2} + (r − Im p_{ϵ})^{2}

≥ \frac{1}{2}(p − 1)^{2} − O(ϵ^{2})(−Im p_{ϵ})^{2} + r^{2} + 2r(−Im p_{ϵ}) + (−Im p_{ϵ})^{2}

≥ r^{2} + \frac{1}{2}(p − 1)^{2} + 2r(−Im p_{ϵ}) + \frac{1}{2}(−Im p_{ϵ})^{2} :

|p_{ϵ} − 1 − ir|^{2} ≥ r^{2} + \frac{1}{2}(p − 1)^{2} + 2r(−Im p_{ϵ}) + \frac{1}{2}(−Im p_{ϵ})^{2},  (12.87)

near K in Σ_{p}^{ϵ_{0}}.

Using these estimates and a variant of (12.81), we shall first prove:

Proposition 12.16 For 0 ≤ r ≤ 1, we have

q(u, u) ≥ (r^{2} − Crh)∥u∥_{H_{Φϵ}}^{2}.  (12.88)

Proof. We drop the subscript H_{Φ_{ϵ}} for the corresponding norms and scalar products. Consider,

q(u, u) − r^{2}∥u∥^{2} = ((P − 1 − ir)u|(P − 1 − ir)u) − r^{2}∥u∥^{2}  (12.89)
= ∥(P − 1)u∥^{2} + ir[((P − 1)u|u) − (u|(P − 1)u)]
≥ r(∥(P − 1)u∥^{2} − 2Im (P u|u))

= r(∫ (|p_{ϵ} − 1|^{2} − 2Im p_{ϵ})|u|^{2}e^{−2Φ_{ϵ}/h}L(dx) + O(h)∥u∥_{H_{Φϵ,m}}^{2}).

From (12.85), (12.86) it is clear that |p_{ϵ} − 1|^{2} − 2Im p_{ϵ} is ≥ 0 in Σ_{p}^{ϵ_{0}} and ≥ O(1)^{−1}⟨ξ⟩^{2} in the complement of Σ_{p}^{ϵ_{0}}. Using this in (12.89), we get q(u, u)− r^{2}∥u∥^{2} ≥ −Crh∥u∥^{2}. #

We want to make finite rank perturbations of q for which the lower bound (12.88) improves. Fix a sufficiently small ϵ > 0 for which the previous estimates hold. We start by combining (12.82), (12.83), (12.85), (12.87) to see that for δ > 0 small enough, we have

|p_{ϵ} − 1 − ir|^{2} ≥ r^{2} + rδ in R^{2n} \ W_{Cδ,r},  (12.90)

where

W_{Cδ,r} = {ρ ∈ R^{2n}; (p(ρ) − 1)^{2} + 2rH_{p}G + (H_{p}G)^{2} < 2Crδ + (Cδ)^{2}}, (12.91)

and C > 0 is sufficiently large and independent of δ. (Recall that ϵ > 0 has now been fixed.)

Using this in (12.81), we get

q(u, u) ≥ (r^{2} + rδ − O(h))∥u∥^{2} − (rδ + O(h))∥1_{W_{Cδ,r}}u∥_{L_{Φ}^{2}}^{2},  (12.92)

where Φ = Φ_{ϵ}.

We shall next show that the last term in (12.92) is very small on certain subspaces of H_{Φ} of finite and explicitly controled codimension. Let χ_{0} ∈ C_{0}^{∞}(B(0, 1)) (where B(0, 1) is the open unit ball in C^{n}) be a radial function with ∫ χ_{0}(x)L(dx) = 1. Let ξ(x) = \frac{2}{i} \frac{∂Φ(x)}{∂x}. Using the mean-value property for the holomorphic function y ↦ e^{i(x−y)·ξ(x)/h}u(y), we get the representation of the identity operator on H_{Φ}:

Iu(x)(= u(x)) = h^{−n} ∫ e^{\frac{i}{h} (x−y)·ξ(x)}χ_{0}(\frac{x − y}{h^{1/2}} )u(y)L(dy).  (12.93)

Let

K(x, y; h) = e^{\frac{i}{h} (x−y)·ξ(x)}χ_{0}(\frac{x − y}{h^{1/2}} )h^{−n}.

Then the reduced kernel e^{−(Φ(x)−Φ(y))/h}K(x, y; h) is O(1)h^{−n}χ_{0}(\frac{x−y}{h^{1/2}} ) and we deduce that the last member of (12.93) defines a bounded operator: L_{Φ}^{2} → L_{Φ}^{2} of norm O(1) uniformly with respect to h.

Let x be close to x̃. We shall compare the kernel K(x, y; h) with the rank 1 kernel

K̃_{x}(x, y; h) = e^{i(x−y)·ξ(_{x})/h}χ_{0}(\frac{x̃ − y}{h^{1/2}} )h^{−n}.

If |x − x̃| ≤ ϵh^{1/2} (with a new ϵ > 0, the earlier one being now fixed and forgotten), we check that the difference betwen the corresponding reduced kernels is O(ϵh^{−n}) and has its support in a domain |x − y| ≤ Const. h^{1/2}.

Now let W ⊂⊂ C^{n} be a compact set which can be covered by M̃(W, ϵh^{1/2}) balls, B_{j} = \overline{B(x_{j}, ϵh^{1/2})}. Let W = ∪_{M} ; j=1W_{j} be a partition of W with W_{j} ⊂ B_{j}, W_{j} ∩ W_{k} = ∅ for j ≠ k. For x ∈ W_{j}, we put x̃(x) = x_{j}. Then W ∋ x ↦ x̃(x) takes at most M̃(W, ϵh^{1/2}) values and |x − x̃(x)| ≤ ϵh^{1/2}. For x ∉ W, we put χu(x) = 0 and for x ∈ W, we put

χu(x) = h^{−n} ∫ e^{\frac{i}{h} (x−y)·ξ(_{x}(x))}χ(\frac{x̃(x) − y}{h^{1/2}} )u(y)L(dy).

Then the discussion above gives:

Lemma 12.17 χ : L_{Φ}^{2} → L_{Φ}^{2} is of finite rank ≤ M̃(W, ϵh^{1/2}) and

∥1_{W} u − χu∥_{L_{Φ}^{2}} ≤ Cϵ∥u∥_{Φ}, u ∈ H_{Φ}(C^{n}).  (12.94)

We next estimate M̃(W, r):

Lemma 12.18 Let W ⊂ C^{n} be compact and let M̃(W, r) be the minimal number of closed balls of radius r > 0, centered at points of W, that cover W. Then

Vol (W)r^{−2n} ≤ Vol (B(0, 1))M̃(W, r) ≤ 2^{2n}Vol (W + B(0, \frac{r}{2}))r^{−2n}. (12.95)

Proof. Clearly

Vol (W) ≤ M̃(W, r)Vol (B(0, r)) = M̃(W, r)Vol (B(0, 1))r^{2n},

which gives the first inequality in (12.95). To prove the second inequality, let A(r) be the maximum number of closed disjoint balls of radius r/2 that can be centered at points of W, and let B(x_{j}, r/2), j = 1, ..., A(r) be such a family of balls. Then,

A(r)Vol (B(0, \frac{r}{2})) ≤ Vol (W + B(0, \frac{r}{2})),

so that

A(r)Vol (B(0, 1)) ≤ 2^{2n}r^{−2n}Vol (W + B(0, \frac{r}{2})).  (12.96)

The second part of (12.95) will then follow, if we prove that M̃(W, r) ≤ A(r), and to do so, it is enough to prove that the \overline{B(x_{j}, r)} cover W. If that were not the case, there would be a point x ∈ W which is not in any of the \overline{B(x_{j}, r)} and hence \overline{B(x, r/2)} ∩ \overline{B(x_{j}, r/2)} = ∅ for every j, which contradicts the maximality of the disjoint family \overline{B(x_{j}, r/2)}. #

Lemma 12.19 Let q̃ ≥ 0 be a C^{1,1}-function on C^{n} with lim inf_{|x|→∞} q̃ > 0. If λ_{0} > 0 is sufficiently small, there exists a constant C_{1} > 0 such that for h ≤ λ ≤ λ_{0}, we have for 0 ≤ ϵ ≤ 1:

Vol (W_{λ} + B(0, ϵh^{1/2})) ≤ Vol (W_{λ+C_{1}ϵh^{1/2}λ^{1/2}}),  (12.97)

where W_{λ} = {x; q̃(x) < λ}.

Proof. We prove the corresponding inclusions for the sets. If x ∈ K_{λ}, we have q̃(x) ≤ λ and |q̃′(x)| ≤ Cλ^{1/2}, so if |y| ≤ ϵh^{1/2}, then

q̃(x + y) ≤ λ + C̃(λ^{1/2}ϵh^{1/2} + ϵ^{2}h) ≤ λ + C_{1}ϵh^{1/2}λ^{1/2}.

#

We now return to (12.92) and apply the last three lemmas with q̃ = (p(ρ) − 1)^{2} + 2rH_{p}G + (H_{p}G)^{2} in (12.91). We conclude that there exists an operator χ_{r,δ} = O(1) : L_{Φ}^{2} → L_{Φ}^{2} of rank

≤ Vol (W_{Cδ,r} + B(0, ϵh^{1/2}))ϵ^{−2n}h^{−n}
≤ Vol (W_{Cδ+C_{2}δ^{1/2}r^{−1/2}ϵh^{1/2},r})ϵ^{−2n}h^{−n},

when rδ ≥ C_{0}h, such that

∥(1_{W_{Cδ,r}} − χ)u∥ ≤ O(ϵ)∥u∥, u ∈ H_{Φ}.

Choose ϵ > 0, so that the last “O(ϵ)” is ≤ 1/3 and use it in (12.92):

q(u, u) ≥ (r^{2} + \frac{2rδ}{3} − O(h))∥u∥^{2} − (rδ + O(h))∥χu∥^{2}, u ∈ H_{Φ,m}. (12.98)

Notice that

rank (χ) ≤ CV (r, Cδ)h^{−n},  (12.99)

for a new sufficiently large constant C > 0.

Proof of Theorem 12.15. Consider still the Hermitian form (12.79):

q(u, u) = ((P − 1 − ir)u|(P − 1 − ir)u)_{H_{Φ}}, u ∈ H_{Φ,m}.

From (12.81) it follows that q is a closed quadratic form with domain H_{Φ,m}. Consequently, by the Lax-Milgram theorem, there exists a unique self-adjoint operator Q ≥ 0 in H_{Φ} such that

D(Q) ⊂ H_{Φ,m},  (12.100)
q(u, u) = (Qu|u), u ∈ D(Q).  (12.101)

Formally, Q = (P −1−ir)^{∗}(P −1−ir). Let μ_{1} ≤ μ_{2} ≤ ... be an enumeration of first all the discrete eigenvalues of Q^{1/2} below the essential spectrum, repeated according to their multiplicity, and then if there are only finitely many such eigenvalues, the infimum of the essential spectrum repeated “ad eternam”. Then we have the max-min formula

μ_{N}^{2} = sup_{E is a subspace of H_{Φ} of codimension ≤N−1} inf_{{u∈E \atop ∥u∥=1}} q(u, u),  (12.102)

with the convention that q(u, u) = +∞ if u ∈ H_{Φ} \ H_{Φ,m}.

Proposition 12.16 tells us that

μ_{1} ≥ \sqrt{r^{2} − Crh} ≥ r − O(h),  (12.103)

where we now restrict all parameters as in Theorem 12.15. On the other hand, (12.98), (12.99) show that if E = N (χ) (the null-space of χ), then

inf_{{u∈E \atop ∥u∥=1}} q(u, u) ≥ r^{2} + \frac{2rδ}{3} − O(h),

and since codim N (χ) ≤ CV (r, Cδ)h^{−n}, we conclude from (12.102), that

μ_{N}^{2} ≥ r^{2} + \frac{2rδ}{3} − O(h), for N ≥ CV (r, Cδ)h^{−n}.  (12.104)

Multiplying δ by a constant (affecting the “C” above), we may assume that,

μ_{N} ≥ r + 2δ, for N ≥ CV (r, Cδ)h^{−n}.  (12.105)

Let z_{1}, z_{2}, ... be the resonances of P in D(1 + ir, r + 2/C_{0}) repeated according to their multiplicity and arranged so that j ↦ |z_{j} − 1 − ir|. Then we have the Weyl inequalities:

μ_{1}μ_{2}...μ_{N} ≤ ∏_{1}^{N} |z_{j} − 1 − ir|,  (12.106)

viewing z_{j} − 1 − ir as the eigenvalues of P − 1 − ir. Notice that these inequalities are “opposite” to the corresponding ones for a compact operator, where the singular values are enumerated in decreasing order and similarly for the moduli of the eigenvalues. See the appendix (dans la prochaine version c’est promis!) for a proof.

Let M(r, δ) be as in the theorem. If M(r, δ) ≤ CV (r, Cδ)h^{−n}, we are done. If not, we apply (12.106) with N = M(r, δ) and get from (12.103), (12.105), with N = [CV (r, Cδ)h^{−n}]:

(r − Ch)^{N}(r + 2δ)^{M−N} ≤ (r + δ)^{M},
(\frac{r + 2δ}{r + δ})^{M} ≤ (\frac{r + 2δ}{r − Ch})^{N},
M log (\frac{r + 2δ}{r + δ}) ≤ N log (\frac{r + 2δ}{r − Ch}).

Here the logarithms are of the order of magnitude δ/r, so we get M ≤ O(1)N. #

By replacing the energy “1” be a real energy E which varies in a neighborhood of E, one obtains a variant of the main theorem. If J ⊂ neigh (1, R) is an interval and 0 < ϵ ≪ 1, let

R(J, ϵ) = {ρ ∈ W; p(ρ) ∈ J, H_{p}G ≤ ϵ}.  (12.107)

Then we have

Theorem 12.20 We make the same assumptions as in Theorem 12.15. Then there are constants C_{0}, C > 0 such that for every subinterval I of [1−ϵ_{0}/2, 1+ ϵ_{0}/2], we have when C_{0}h ≤ δ ≤ C_{0}^{−1},

M_{3}(I, δ) ≤ CVol (R(I + [−Cδ̃, Cδ̃], Cδ))h^{−n},  (12.108)

where δ̃ = max(δ, h^{1/2}). Here M_{3}(I, δ) denotes the number of resonances in the rectangle I − i[0, δ[.

In a later version of these notes we will put in the detailed proof, but in this version we only give the outline: Cover I − i[0, δ[ by a “minimal” number of discs D(E_{j} + ir, r + 2δ) and remember the constraints 1 ≥ r ≥ δ, rδ ≥ C_{0}h. If δ ≥ h^{1/2}, we take r ∼ δ and if C_{0}h ≤ δ ≤ h^{1/2}, we take r ∼ h/δ. Then estimate the number of resonances in D(E_{j} + ir, r + 2δ) by CVol ({ρ ∈ W; p − iH_{p}G ∈ D(E_{j} + ir, r + Cδ)}. When summing over j we notice that every point in the lower half plane can belong to at most a fixed finite number of the discs D(E_{j} + it, r + Cδ). #

12.7 Quick review of the case when the classical flow is hyperbolic

Recall that K, Γ_{±} were defined in Σ_{p}^{ϵ_{0}} for some small fixed ϵ_{0} > 0. Let K̂, Γ̂_{±} be the corresponding sets in Σ_{p}^{2ϵ_{0}}. We make the following hyperbolicity assumptions about the classical dynamics:

(Hyp 1) In a neighborhood Ω_{ρ_{0}} of every point ρ_{0} ∈ K, we can represent Γ̂_{±} as a union of closed disjoint C^{1}-manifolds of dimension n + 1, such that if ρ ∈ Ω_{ρ_{0}} ∩ Γ̂_{+} and if E_{ρ}^{+} = T_{ρ}(Γ̂_{+,ρ}), where Γ̂_{+,ρ} is the corresponding leaf, then E_{ρ}^{+} depends continuously on ρ ∈ Ω_{ρ_{0}} ∩ Γ̂_{+}, and contains H_{p}(ρ). Same assumption about E_{ρ}^{−} = T_{ρ}(Γ̂_{−,ρ}).

(Hyp 2) E_{ρ}^{+} and E_{ρ}^{−} are transversal for every ρ ∈ K̂, and are indpendent of the choice of Ω_{ρ_{0}}, containing ρ.

(Hyp 3) There exists a constant C > 0, such that with Φ_{t} = exp tH_{p}:

∥dΦ_{t}(ν)∥ ≤ Ce^{−t/C}∥ν∥, ν ∈ T_{ρ}(R^{2n})/E_{ρ}^{+}, ρ ∈ K̂, t ≥ 0,  (12.109)
∥dΦ_{−t}(ν)∥ ≤ Ce^{−t/C}∥ν∥, ν ∈ T_{ρ}(R^{2n})/E_{ρ}^{−}, ρ ∈ K̂, t ≥ 0,  (12.110)

where dΦ_{t} is considered as a map T_{ρ}(R^{2n})/E_{ρ}^{±} → T_{Φ_{t}(ρ)}(R^{2n})/E_{Φ_{t}(ρ)}^{±}.

Theorem 12.21 Under the above assumptions and after an arbitrarily small decrease of ϵ_{0}, we can find G as in Theorem 12.15 such that

H_{p}G ≥ \frac{1}{C} dist (·, K̂)^{2}, near K in Σ_{p}^{ϵ_{0}} and > 0 in Σ_{p}^{ϵ_{0}} \ K.  (12.111)

Let L ⊂ R^{2n} be bounded and put L_{ϵ} = {ρ ∈ R^{2n}; dist (ρ, L) < ϵ}, for ϵ > 0. We define the Minkowski codimension of L; codim (L), to be the supremum of all d ≥ 0, such that Vol (L_{ϵ}) ≤ O(ϵ^{d}) when ϵ → 0. Clearly 0 ≤ codim (L) ≤ 2n when L ≠ ∅. We say that L is of pure dimension if the supremum is attained, so that Vol (L_{ϵ}) ≤ O(ϵ^{codim (L)}). Put dim (L) = 2n − codim (L). The following result is a consequence of Theorem 12.20 and Theorem 12.21:

Theorem 12.22 Under the above assumptions, let d = codim (K̂) if K̂ is of pure codimension, and otherwise let 0 ≤ d < codim K̂. Then there is a constant C_{0} > 0 such that for 0 < h ≤ C_{0}^{−1}, C_{0}h ≤ δ ≤ C_{0}^{−1}, the number of resonances of P in the rectangle ] − \frac{ϵ_{0}}{2} , \frac{ϵ_{0}}{2} [−i[0, δ[ is ≤ C_{0}δ^{d/2}h^{−n}.

Remains to add to this section: The proof of Theorem 12.21 and a discussion of the example of several strictly convex obstacles and the corresponding example of approximating potentials.

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