LOWER BOUNDS FOR SHAPE RESONANCES WIDTHS OF LONG RANGE SCHRÖDINGER OPERATORS

by

Nicolas Burq

Abstract. — In this article we give general lower bounds of shape resonance widths in the semi-classical limit for Schrödinger operators in the exterior of smooth compact obstacles with Dirichlet or Neuman boundary conditions and with long range dilation analytic potentials. These lower bounds are exponentially small with respect to the Planck constant. Résumé. — On donne dans cet article des bornes inférieures universelles, en limite semiclassique, de la taille des résonances de forme associées aux l’opérateurs de Schrödinger à l’exterieur d’obstacles avec des conditions au bord de Dirichlet ou de Neumann et des potentiels à longue portée analytiquement dilatables. Ces bornes inférieures sont exponentiellement petites par rapport à la constante de Planck.

1. Introduction, hypotheses

The purpose of this article is the study of the localization of resonances for the semiclassical Schrödinger operator near positive energy levels. The shape resonances are the metastable states of a system whose evolution is described by a Hamiltonian, H, depending upon the Planck constant h. An important feature of the resonances is their lifetime (the reverse of their width) which describes the time needed by a wave packet associated to the resonance and initially localized in a compact region of the space, to develop a non zero probality density in the outside region. In this paper we shall study the semi-classical approximation when the planck constant tends to 0. Most results about the shape resonances are concerned about the existence of resonances and upper bounds for their width (see the works by J.M. Combes, P. Duclos, M. Klein and R. Seiler [CDKS87], P.D. Hislop and I.M. Sigal [HS89] and by B. Helffer and J. Sjöstrand [HS86]) However, in the case of trapped energy levels, there are very few results about non existence of resonances or more generally about lower bounds for the width of shape resonances. To our knowledge, the only known results about such matters were particular cases: for a Schrödinger operator in space dimension 1 equal to −∂_{x}^{2} + V (x) with a potential compactly supported E.M. Harrel [Har82] gave lower bounds. In space dimension d ≥ 1, B. Helffer and J. Sjöstrand [HS86] for analytic dilation potential with a unique non degenerate local minimum and energy near the bottom of the well gave a complete asymptotic expansion of the resonances. Later on C.Fernandez and R.Lavine [FL90] extended the results by E.Harrel to the case of space dimension d ≥ 1 and potential with compact support. Finally the author [Bur98] has given such lower bounds for the Helmoltz operator (null potential) outside any obstacle (whose complementary is connected), with Dirichlet conditions on the boundary. These results have been recently extended to superexponentially decaying perturbations of the Laplace operator (O(e^{−|x|^{2+ε}}) by G. Vodev [Vod00], using the same kind of methods. In fact, the methods developped in [Bur98] could apply to compactly supported perturbations of the Laplace operator (in the semi-classical framework).

The result we obtain in this article is the following:

Theorem 1. — Consider a long range (O (1/|x|^{β}) , β > 0) self adjoint analytic dilation perturbation of the semi-classical Laplace operator −h^{2}Δ (satisfying the assumptions [1..4] of section 2). Then for any 0 < E_{m} < E_{M} < +∞, there exists C, d, h_{0} > 0 such that for any 0 < h < h_{0}, the operator P (h) has no resonance in the set

{z ∈ ℂ; dist (z, [E_{m}, E_{M}]) ≤ Ce^{−d/h} (1.1)

Remark 1.1. — It is very likely that our method could deal with unbounded energy level (e.g. to control the constants with respect to E_{M}). However, the question of the null energy is very different and probably deserves different methods.

The result of theorem 1 means that near any strictly positive energy E, the resonances can not accumulate to the real axis faster than exponentially. In fact, we shall obtain a more precise result: the resonances are defined using complex scaling as the eigenvalues of certain non self adjoint operator. We shall give an (exponentially large) estimation of the norm of the resolvent of this operator on any compact K ⊂ ℝ_{+}^{∗}.

A natural question would be to determine the optimal constant d near a fixed level of energy (e.g. for K a small neighbourhood of a point E) in terms of the geometry of the problem. In the one dimensional case or if the potential and the metric are rotation invariant, it is possible to recover, following our method, some classical results and to express C in terms of the Agmon distance. However, in the general case, the study of the Helmoltz resonator and results by A. Martinez on the tunneling effect for the double well [Mar88], section 7, make us think that the optimal constant is more likely to be a microlocal quantity than a local one.

The starting point for the proof of this theorem is the same as in [Bur98]: use Carleman estimates to deal with the region where one has no information (any compact region). Then in [Bur98] it was possible to conclude using that the operator was equal to −h^{2}Δ outside of a compact set, by making computations on the unperturbed outgoing resolvent of the Laplace operator. Here we shall use the fact that the operator P tends to −h^{2}Δ when r tends to +∞ in order to go back from Carleman estimates to Mourre type estimates (positive commutators estimates) and then perform analytic dilation to conclude. These two steps rely heavily on explicit and quite technical computations. We were unfortunately unable to give a more conceptual proof of the result.

Remark 1.2. — It is likelly that the result might also hold without the assumption that the perturbation tends to zero at infinity, provided that one assumes the existence of certain escape functions. For the sake of simplicity we did not want to get into such refinments.

Remark 1.3. — The dilation analyticity assumption is necessary to define the resonances. However, dropping this assumption and replacing it by a symbol-like decay for the perturbation, the methods developped in this paper (and the limit absorption principle) would lead to a weighted estimate for the outgoing resolvent. To keep this paper a reasonable length, we leave this for a forthcoming article.

The paper is organized as follows: in part 2 we give some basic facts about complex scaling. These results are due to J. Sjöstrand and M. Zworski. We also state a result (theorem 3) about the resolvent of the dilated operators which implies theorem 1. In part 3 we recall some results on Carleman estimates for semi-classical second order boundary value problems. In the context of the Laplace operator, these results are due to G. Lebeau and L. Robbiano [LR95b, LR95a]. Here we state the results for Helmoltz-like operators and for the convenience of the reader we outline the proofs. In the most calculatory part of the paper (part 4) we construct the phase functions that we shall use in part 6 and 7 to prove the main technical result of the article (propositions 6.4 and 7.4. In part 5 we show how theorem 3 and 4 can be deduced from these technical propositions (and from Carleman estimates). In section 8, we give a byproduct of this work concerning the decay of the local energy for solutions to the wave equation with asymptotically flat metric (announced below as theorem 2). Finally ,in an appendix we give an outline of some technical results used to simplify the form of our operators. Almost all the constants that appear in the paper are fixed in section 4 by the construction of these phase functions. This is the reason why we had to write this section at this point of the paper. However, for a first lecture, we strongly recommend the reader to postpone the analysis of the proofs of the results in section 4.

To finish this introduction let us try to give a flavour of some consequences of our results for solutions of wave equation that we shall developp in section 8.

Consider Θ ⊂ ℝ^{d} a (possibly empty) smooth obstacle whose complementary, Ω = Θ^{c}, is connected. Suppose that the boundary of Θ, ∂Θ is the disjoint union of two components, Γ_{D} and Γ_{N}. Note g a smooth (positive definite) metric g = (g_{i,j})_{i,j=1,...d} in Ω, satisfying the smoothness, analyticity at infinity, ellipticity and asymptotic flatness assumptions of section 2. Consider for (u_{0}, u_{1}) ∈ C_{0}^{∞} (Ω), u (t) the solution of the wave equation:

(1.2)  (∂_{t}^{2} − \frac{1}{\sqrt{detg}} ∂_{x_{i}} \sqrt{detg}g^{i,j} (x) ∂_{x_{j}}) u = 0
(1.3)  u |_{Γ_{D}}= 0  \frac{∂u}{∂n} |_{Γ_{N}} +b (x) u |_{Γ_{N}} = 0
(1.4)  u |_{t=0}= u_{0}  ∂_{t}u |_{t=0}= u_{1}

with ∂_{n} the normal derivative associated to the metric g and (g^{i,j}) = (g_{i,j})^{−1} and b(x) ∈ ℝ (to have a self adjoint operator).

Theorem 2. — For any k ∈ ℝ^{+} and any R > 0 there exists a constant C > 0 such that for any initial data, (u_{0}, u_{1}) ∈ C_{0}^{∞} (Ω)^{2} with support in the ball Ω ∩ B (0, R) and any t > 1 we have

∥u (t) ∥_{H^{1}(Ω∩B(0,R))} + ∥∂_{t}u (t) ∥_{L^{2}(Ω∩B(0,R))} ≤ \frac{C}{(log t)^{k}} [∥u_{0}∥_{H^{1+k}} + ∥u_{1}∥_{H^{k}}] (1.5)

There exists R_{0} > 0 such that for any R > R_{0} there exists a constant C > 0 such that for any initial data, (u_{0}, u_{1}) ∈ H^{1}×L^{2} with support in the ring Ω∩(B (0, R) \ B (0, R_{0})) we have

(1.6) ∫_{0}^{+∞} ∥u (t) ∥_{H^{1}(Ω∩(B(0,R)\B(0,R_{0})))}^{2} + ∥∂_{t}u (t) ∥_{L^{2}(Ω∩(B(0,R)\B(0,R_{0})))}^{2}

≤ C [∥u_{0}∥_{H^{1}}^{2} + ∥u_{1}∥_{L^{2}}^{2}]

2. A review of complex scaling in the semi-classical case

2.1. Assumptions. — Let us first precise the framework of this article:

Let Θ ⊂ ℝ^{d} be a (possibly empty) smooth obstacle whose complementary, Ω = Θ^{c}, is connected. Suppose that the boundary of Θ, ∂Θ is the disjoint union of two components, Γ_{D} and Γ_{N}. Consider a semi-classical self adjoint operator defined on L^{2} (Ω):

Q = ∑_{i,j=1}^{d} a_{i,j}(x)hD_{x_{i}}hD_{x_{j}} + ∑_{i=1}^{d} (b_{i} (x, h)) hD_{x_{i}} + V (x, h) , D_{x} = \frac{∂_{x}}{i} (2.1)

with domain

D = \{u ∈ H^{2} (Ω) ; u |_{Γ_{D}}= 0 and \frac{∂u}{∂n} |_{Γ_{N}} +b (x) u |_{Γ_{N}} = 0\} (2.2)

where ∂_{n} is the normal to the boundary with respect to the metric g = (a_{i,j})^{−1}. Observe that Γ_{D} or Γ_{N} or the both of them can be empty and that the self adjointness of the operator Q imposes compatibility conditions for the Neumann part of the boundary condition.

Following J. Sjöstrand [Sjö97], we make the following assumptions about the operator Q 1. (Smoothness.) The functions a_{i,j}, b_{i}, V and their derivatives are (uniformly with

respect to 1 > h > 0) bounded for the C^{∞} (ℝ^{d}) topology (in any compact set)

2. (Analyticity at infinity.) There exists θ_{0}, ε > 0 and R_{0} > 0 such that the functions a_{i,j}, a_{i}, b_{i}, V and W are analytic in x in the domain

(2.3) Λ_{θ_{0},R_{0},d_{0}} = {rω; ω ∈ ℂ, dist (ω, 𝕊^{d−1}) < d_{0},

r ∈ ℂ, |r| > R_{0}, arg (r) ∈ [−θ_{0}, θ_{0}]}

3. (Ellipticity.) The operator Q is supposed to be elliptic

∃C > 0; ∀x ∈ Ω, ∀ξ ∈ ℝ^{d} ∑_{i,j=1}^{d} a_{i,j} (x) ξ_{i}ξ_{j} ≥ C|ξ|^{2} (2.4)

4. (Asymptotic flatness.) The operator Q is close to −h^{2}Δ at infinity:

\begin{matrix}∃β > 0, M > 0; ∀x ∈ Λ_{θ_{0},R_{0},d_{0}}, ∀0 < h < 1, \\ |a_{i,j} (x) − δ_{i,j}| ≤ \frac{M}{|x|^{β}} , |b_{i} (x, h) | + |V | (x, h) ≤ \frac{M}{|x|^{β}} \end{matrix} (2.5)

Remark 2.1. — According to Cauchy formula, (2.5) imply (in Λ_{θ_{0}/2,R_{0}+1,d_{0}/2})

|∂_{x}^{α}a_{i,j} (x) | + |∂_{x}^{α}b_{i} (x) | + |∂_{x}^{α}V | (x) ≤ \frac{M}{|x|^{|α|+β}} (2.6)

We shall note

\begin{matrix}Q_{0} (x, D_{x}) = ∑_{i,j=1}^{d} a_{i,j} (x) hD_{x_{i}}dD_{x_{j}} \\ q (x, ξ, h) = ∑_{i,j=1}^{d} a_{i,j} (x) ξ_{i}ξ_{j} + ∑_{i=1}^{d} b_{i} (x, h) ξ^{i} \end{matrix} (2.7)

It was shown by J. Sjöstrand [Sjö97] that with these hypotheses one can define the resonances of the semi classical operator Q in any sector S_{θ_{0}} = {z ∈ ℂ; z = e^{−iθ}r, r ≥ 0, θ ∈ [0, θ_{0}]} and that these resonances form a discrete set. However, before performing the analytic dilation, we shall make some reductions concerning the form of the operator Q.

2.2. Reductions for the operator Q. — The proof of the following result is given in appendix A:

Proposition 2.2. — There exists R_{0} > 0, θ_{1} > 0 and a holomorphic change of coordinates:

χ : R_{0}+]0, +∞[e^{i]−θ_{0},θ_{0}[} × B(𝕊^{d−1}, ε) → Ω ⊂ ℂ^{d} (2.8)

(where B(𝕊^{d−1}, ε) is an ε-neighborhood of 𝕊^{d−1} in the complexified d − 1 dimensionnal sphere) satisfying the following:

1. χ induces a diffeomorphism from B(0, R_{0})^{c} to Ω ∩ ℝ^{d}

2. Seen as a map from ℂ^{d} to ℂ^{d}, χ is asymptotically equal to the identity: Note χ_{r} and χ_{σ} the components of χ in radial coordinates. Then

∥χ_{σ}(r, σ) − σ∥ ≤ \frac{ϵ}{|r|^{β}} (2.9)

and

∥\frac{χ_{r}(r, σ) − r}{r} ∥ ≤ \frac{ϵ}{|r|^{β}} (2.10)

where ϵ can be taken small if R_{0} is large. and such that in this new coordinate system the operator Q still has the form (2.1), satisfies the analyticity and asymptotic flatness (with possibly different constants θ_{0}, d_{0} and M) assumption and furthermore its second order part is equal to (in polar coordinates, (r, σ))

−(h∂_{r})^{2} − h^{2}P_{2}(r, σ, \frac{h∇_{σ}}{r}) (2.11)

with P_{2} a quadratic form. In this coordinate system, the operator Q is self adjoint for the measure μ = f(r, σ)drdL, image of the Lebesgue measure on ℝ^{d}. (dL is the Lebesgue measure on 𝕊^{d−1} and f ∼ r^{d−1})

Remark 2.3. — This result is only a global (and holomorphic) version at infinity of a well known result saying that the terms ∂_{r}∂_{σ} can be eliminated by the choice of a geodesic coordinate system. The holomorphic version is straightforward by solving the geodesic equations in the complex space; and the globality at infinity follows from the fact that the operator is asymptotically flat.

We’ll also obtain in appendix A the following result:

Proposition 2.4. — There exists a holomorphic function k = f^{1/2}e^{−f̂/h} with f̂ |_{ℝ^{d}}∈ iℝ such that the conjugate operator P = k^{−1}Qk is self adjoint for the measure drdL and has the following form (for |x| ≥ R_{0}):

P = −h^{2}∂_{r}^{2} − h^{2} \frac{Δ_{σ}}{r^{2}} + q_{2}(r, σ, \frac{h∇_{σ}}{r}) + q_{1}(r, σ, \frac{h∇_{σ}}{r} , h) + V (r, σ, h) (2.12)

where Δ_{σ} is (in the σ coordinate system) the Laplace operator on the sphere 𝕊^{d−1} and q_{2}, q_{1} and V are self adjoint operators (of orders 2, 1 and 0 respectively) such that if

q_{2}(r, σ, \frac{h∇_{σ}}{r}) = ∑_{i,j} q_{2}^{i,j}(r, σ)\frac{h∂_{σ_{i}}}{r} \frac{h∂_{σ_{i}}}{r} (2.13)

q_{1}(r, σ, \frac{h∇_{σ}}{r} , h) = ∑_{k} q_{1}^{k}(r, σ, h)\frac{h∂_{σ_{i}}}{r} (2.14)

then the coefficients q_{2}^{i,j}, q_{1}^{k} and V are holomorphic in Λ_{θ′_{0},R′_{0},d′_{0}} and satisfy there the following estimates (uniformly with respect to 0 < h < 1)

|q_{2}^{i,j}(r, σ)| + |q_{1}^{k}(r, σ, h)| + |V (r, σ, h)| ≤ \frac{M}{|r|^{β}} . (2.15)

Remark 2.5. — In the first part of this proposition, the conjugation by f^{1/2} makes the operator self adjoint with respect to the metric drdL and the conjugation by e^{f̂/h} does eliminate the first order term in ∂_{r}. The fact that f̂ is purely imaginary is a simple consequence of the fact that this first order term is self adjoint (hence the coefficient of ∂_{r} is purely imaginary). The second part of the proposition is simply the traduction that the operator is (taking the conjugation into account) close to the operator

P_{0} = −h^{2}∂_{r}^{2} − h^{2} \frac{Δ_{𝕊^{d−1}}}{r^{2}} (2.16)

Remark 2.6. — Since the functions q_{2}^{i,j}, q_{1}^{k} and q_{0} are holomorphic in conic domains, the Cauchy formula imply (after disminishing the angle of these domains), that we have for any α ∈ ℕ^{d},

\begin{aligned}|∂_{r}^{α_{1}}∂_{σ}^{α_{2}} (q_{2}^{i,j}) (r, σ)| & ≤ \frac{M}{|r|^{β+|α_{1}|}} , \\ |∂_{r}^{α_{1}}∂_{σ}^{α_{2}} (q_{1}^{k}) (r, σ, h)| & ≤ \frac{M}{|r|^{β+|α_{1}|}} , \\ |∂_{r}^{α_{1}}∂_{σ}^{α_{2}} (V) (r, σ, h)| & ≤ \frac{M}{|r|^{β+|α_{1}|}} \end{aligned} (2.17)

Here, we work in a (finite number of) local coordinate system on the sphere, but for the sake of simplicity, we shall do as if this coordinate system was global.

Remark 2.7. — In the sequel we’ll always work in the coordinate system above and we’ll always perform the conjugation of proposition 2.4. In fact in the set Ω ∩ B(0, R_{0}) where this special coordinate system does not exist, the analysis we use to treat what happens (Carleman estimates) is invariant by change of coordinates. So, for our purpose, the fact that this coordinate system does not exist everywhere will be transparent.

2.3. Analytic dilation. — We now recall the definition of resonances by Sjöstrand [Sjö97] in this framework. We follow very closely [Sjö97].

For given ε_{0} > 0, B >> 1, we can construct a smooth function f_{θ} (t) : [0, π] × [0, +∞[→ ℂ, injective for any θ with the following properties:

(i) f_{θ} (t) = t for 0 ≤ t ≤ B (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} depends only on ε_{0} and B.

(v) arg(f_{θ} (t)) is an increasing function of θ and t. (vi) arg(f_{θ} (t)) > 0 for t > B.

For technical reasons, we’ll suppose that the function f_{θ}(t) satisfies the following additional assumptions (vii) f_{θ}(t) = te^{iκ_{θ}(t)} with κ_{θ} a real, increasing function such that

|κ′_{θ}|^{2} + |κ′′_{θ}|^{2} + |κ′′′_{θ}|^{2} + |κ_{θ}^{(4)}|^{2} ≤ C_{θ}κ_{θ} (2.18)

The analytic dilation will be performed for θ ∈ [0, θ_{0}]. We will also suppose that the angle θ_{0} > 0 is taken so small that

−Im (\frac{e^{−iκ_{θ_{0}}(t)}}{(1 + itκ′_{θ_{0}}(t))^{2}} ) ≥ \frac{3}{4}(κ_{θ_{0}} + rκ′_{θ_{0}})(t), Im(e^{iκ_{θ_{0}}(t)}) ≥ \frac{3}{4}κ_{θ_{0}}(t) (2.19)

Consider the map

κ_{θ_{0}} : ℝ ∋ x = tω ↦ f_{θ_{0}} (t) ω ∈ ℂ, t = |x|. (ω ∈ 𝕊^{d−1}) (2.20)

Then the image of κ_{θ_{0}}, Γ_{θ_{0}}, is a totally real manifold which coincides with ℝ^{d} along B (0, B).

Define

L_{θ_{0}}^{2} = L^{2}(Γ_{θ_{0}} \ Θ) (2.21)

H_{0,θ_{0}} = {u ∈ H^{1} (Γ_{θ_{0}}) ; u |_{Γ_{D}}= 0} (2.22)

D_{θ_{0}} = \{u ∈ H^{2} (Γ_{θ_{0}}) ; u |_{Γ_{D}}= 0, (\frac{∂u}{∂n} + b(x)u) |_{Γ_{N}} = 0 (2.23)

where H^{2} (Γ_{θ_{0}}) and H^{1} (Γ_{θ_{0}}) are equipped with their natural semi-classical norms:

\begin{matrix}∥u∥_{H_{0,θ_{0}}}^{2} = ∥u∥_{L^{2}}^{2} + ∥h∇u∥_{L^{2}}^{2} \\ ∥u∥_{D_{θ_{0}}}^{2} = ∥u∥_{L^{2}}^{2} + ∥h∇u∥_{L^{2}}^{2} + ∑_{i,j} ∥hD_{i}hD_{j}u∥_{L^{2}}^{2} \end{matrix} (2.24)

Lets consider the operator P_{θ_{0}} (h) acting on L^{2} (Γ_{θ_{0}} \ Θ) with domain D_{θ_{0}} which inside B (0, B) coincide with P (h) and outside is (in the polar coordinates coming from κ_{θ_{0}}) equal to

P_{θ_{0}} (h) = P (f_{θ_{0}} (t) ω, f′_{θ_{0}} (t) D_{t}, D_{ω}) (2.25)

The following lemma is shown in [Sjö97] (lemma 5.1 and 5.2)

Lemma 2.8. — If z ∈ ℂ \ {0}, arg (z) ≠ −2θ_{0}, then the operator P_{θ_{0}} − z from D_{θ_{0}} to L_{θ_{0}}^{2} is a Fredholm operator of index 0. Furthermore, if arg (z) < −2θ_{1} ≤ −2θ_{2} then

(2.26) dim Ker (P_{θ_{1}} − z) = dim Ker (P_{θ_{2}} − z)

This lemma and the analytic Fredholm theory show that the spectrum of P_{θ} in ℂ \ e^{−2iθ}[0, +∞[ is discrete and included in ] − ∞, 0] ∪ e^{−2i[0,θ[}[0, +∞[, the part in ] − ∞, 0] beeing the eigenvalues of the operator P (h).

Definition 2.9. — We say that z ∈ e^{−2i[0,θ[}[0, +∞[ is a resonance for the operator P (h) if and only if z ∈ σ (P_{θ}) for some (and hence all) θ > −arg (z) /2.

Since the operator P_{θ} is a Fredholm operator of index 0 it is equivalent to define the resonances as the eigenvalues of the operator P_{θ} (h). The theorem 1 is now a straightforward consequence of the following result:

Theorem 3. — For any compact set, K ⊂ ℝ_{+}^{∗} there exists B and θ_{0} > 0, C, C′, h_{0} > 0 such that for any 0 < h < h_{0} and any e ∈ K, the operator P_{θ_{0}} (h) − eId is invertible on L^{2} (Γ_{θ_{0}}) and one has

∥ (P_{θ_{0}} (h) − eId)^{−1} ∥_{L^{2}(Γ_{θ_{0}})} ≤ Ce^{C′/h} (2.27)

Remark 2.10. — We are going to prove theorem 3 for some particular choices of B and of the function f_{θ}. However it is a consequence of the works by J. Sjöstrand and M. Zworski [SZ91] (see the proof of lemma 3.4) that theorem 3 (i.e. the estimate (2.27) for a particular choice of B and θ_{0} implies that the estimate (2.27) is still valid (with a different constant C) for any choice of B′ > B and 0 < θ′ < θ_{0}, satisfying assumptions (i) to (vi). Even more, we could perform the analytic dilation in the initial coordinate system in ℝ^{d} (instead of doing it in the coordinate system of appendix A) and estimate (2.27) would still be valid.

As a by-product of the proof of theorem 3 we obtain the following:

Theorem 4. — There exists R_{1} such that for any R_{2} > 0 and any compact set, K ⊂ ℝ_{+}^{∗} there exists C, h_{0} > 0 such that for any 0 < h < h_{0} and any e ∈ K one has

∥1_{R_{1}≤|x|≤R_{2}} (P (h) − eId)^{−1} 1_{R_{1}≤|x|≤R_{2}}∥_{L^{2}} ≤ \frac{C}{h} (2.28)

where (P (h) − eId)^{−1} denotes the outgoing resolvent of the operator P.

Remark 2.11. — The C/h bound in estimate (2.28) is the natural bound obtained in non trapping cases for the resolvent truncated in balls. Here we see that if the cuttoff is performed in rings, then the natural estimate is always valid. This result is a weak (but very general) version of a recent result obtained by Martinez in a particular case showing that in the case of a potential with a unique non degenerate minimum, even if there are resonances exponentially close to the real axis, at these points the residue of the scattering matrix is exponentially small; these two exponentials compensating exactly each other.

Remark 2.12. — By a simple scaling argument, we can suppose that the compact set, K is equal to [1, 2].

3. A review of Carleman estimates

In this section we give some Carleman estimates up to the boundary for the Schrödinger operator. Such kind of estimates were originaly proved by Hormander in the interior, then extended by G. Lebeau and L. Robbiano for the Laplace operator with Dirichlet ([LR95a]) and Neumann ([LR95b]) conditions. The proof of these estimates for the Schrödinger operator is essentially the same as the proof by Lebeau and Robbiano.

Let Ω be a smooth, bounded domain of ℝ^{d}. Let e ∈ [1, 2], P^{e} = P (h) − e be the Schrödinger operator with semi-classical principal symbol p^{e}(x, ξ, h).

p^{e}(x, ξ, h) = ∑_{i,j} a_{i,j}(x)ξ_{i}ξ_{j} + ∑_{i} b_{i}(x, h)ξ_{i} + V (x, h) − e (3.1)

Consider a function φ ∈ C^{∞}(ℝ^{d}) which satisfies the hypoellipticity assumptions (uniformly with respect to 0 < h < h_{0}):

∃c > 0; ∀x ∈ \overline{Ω} ∀ξ ∈ ℝ^{d}, p_{φ}^{e}(x, ξ) = 0 ⇒ \{Re p_{φ}^{e}, Im p_{φ}^{e}\} ≥ c (3.2)

where {f, g} = ∑_{i} ∂_{ξ_{i}}f∂_{x_{i}}g − ∂_{x_{i}}f∂_{ξ_{i}}g is the Poisson bracket of the two functions f and g. We shall also suppose that

\frac{∂φ}{∂n} |_{∂Ω}≠ 0 and ∇φ ≠ 0 in \overline{Ω}. (3.3)

The operator P_{φ}^{e} is defined by P_{φ}^{e} = e^{φ/h}P^{e}e^{−φ/h}. His principal symbol is

p_{φ}^{e} = p^{e}(x, ξ + iφ′_{x}, h). (3.4)

If the function f satisfies the boundary conditions (2.2), then the conjugate function g = e^{φ/h}u satisfies the following:

g |_{Γ_{D}}= 0, (h\frac{∂g}{i∂n} + ihbg + i∇φ · ∂_{n}@g) |_{Γ_{N}} = 0 (3.5)

that we’ll note (with k = −(i∇φ · ∂_{n} + ihb),

g |_{Γ_{D}}= 0, h\frac{∂g}{i∂n} |_{Γ_{N}} = kg |_{Γ_{N}} . (3.6)

Proposition 3.1. — Let φ satisfy conditions (3.2) and (3.3). There exists C > 0, h_{1} > 0, such that for any 0 < h < h_{1} and any g ∈ C^{∞} (\overline{Ω})

(3.7) ∫_{Ω} |P_{φ}^{e}(g)|^{2} + h ∫_{∂Ω} (|g |_{∂Ω} |^{2} + |h∂_{x′}g |_{∂Ω} |^{2} + |h∂_{x_{n}}g |_{∂Ω} |^{2})

≥ Ch ∫_{Ω} (|g|^{2} + |h∇g|^{2}) .

Proposition 3.2. — Let φ satisfy conditions (3.2) and (3.3). Note Γ = Γ_{D} ∪ Γ_{N} ⊂ ∂Ω a union of connected components of the boundary ∂Ω. Suppose that \frac{∂φ}{∂n} > 0 on Γ (\frac{∂}{∂n} is the incoming normal unit on the boundary), then there exists c > 0, h_{1} > 0, such that for any 0 < h < h_{1} and any g ∈ C^{∞} (\overline{Ω}) such that g |_{Γ_{D}}= 0, and (hD_{n}f |_{Γ_{N}} = kf |_{Γ_{N}},

(3.8) ∫_{Ω} |P_{φ}^{e} (g) |^{2} + h ∫_{∂Ω\Γ} (|g |_{∂Ω} |^{2} + |h∂_{x′}g |_{∂Ω} |^{2} + |h∂_{x_{n}}g |_{∂Ω} |^{2})

≥ Ch ∫_{Ω} (|g|^{2} + |h∇g|^{2}) .

The analogues of these propositions are proved by G. Lebeau and L. Robbiano [LR95a, LR95b], for the Laplace operator. We shall give here an outline of the proof in our particular case.

Remark 3.3. — Estimations such as (3.7) and (3.8) are stated globally. However, by means of partitions of unity, it is clearly possible to reduce the proof to local proofs (provided that the phase φ is globally defined). They also depend clearly only on the (semi classical) principal part of the operator. Hence the conjugation by any function h-independant is of no effect.

In the interior such estimations are classical. Hence, for the proof of propositions 3.1 and 3.2, we can assume that the function g is supported by a small neighbourhood of the boundary (remark that in any case the result near the boundary implies the result in the interior). Taking a geodesic coordinate system for the second order elliptic operator P_{0}, component, we can assume also that Ω = {(x′, x_{d}) ∈ ℝ^{d}; x_{d} > 0}, g ∈ C_{0}^{∞} (K), the set of functions C^{∞} up to the boundary of ℝ_{+}^{d}, supported by K = {x ∈ ℝ^{d}; |x| ≤ D} and that P − e is equal to

P − e = h^{2}D_{x_{d}}^{2} + R(x_{d}, x′, h\frac{Dx′}{i}) + 2ahD_{x_{d}}, (3.9)

with R a tangential differential operator of order 2. Conjugating the operator P by a suitable function, we also can assume that the operator considered is self adjoint with respect to the measure dx_{d}dx′. In the sequel, we shall dropp the e exponent and suppose that e is fixed.

Since the operator P is self adjoint, a (x, h) ∈ ℝ, we can then perform a conjugation with e^{ig(x_{d},x′,h)}, \frac{dg}{dx_{d}} = a (x_{d}, x′, h) the h∂_{x_{d}} component, leading to an operator

P − e = h^{2}D_{x_{d}}^{2} + R(x_{d}, x′, h\frac{Dx′}{i}) (3.10)

with a possibly different R. Remark that we conjugate the operator by an h-dependent function; however, g is uniformly bounded with respect to h as well as any derivative and everything works as if g was h-independant.

We recall now how G. Lebeau et L. Robbiano proceed to prove estimations (3.7) and (3.8) for operators of the form (3.10). The technical details are in [LR95a], part 3, pages 347 to 351 and [LR95b] part 5.

We shall denote by S^{j} the space of C^{∞} tangential symbols a (x, ξ′, h) defined in the boundary of K et ξ′ ∈ ℝ^{d−1}, such that for 0 < h < h_{0}

∀α, β, ∃C ∀x, ξ′, h, |∂_{x}^{α}∂_{ξ′}^{β}a| ≤ (1 + |ξ′|)^{j−|β|} (3.11)

Let S_{cl}^{j} ⊂ S^{j} be the subspace of the symbols of the form a (x, ξ, h) = a_{0} (x, ξ) + hb (x, ξ, h) with b ∈ S^{j−1}. Let E^{j} (resp. E_{cl}^{j}) be the space of tangential pseudo-differential operators, A = Op (a) ; a ∈ S^{j} (resp. a ∈ S_{cl}^{j}) defined by

Op (a) u (x_{d}, x′) = (2πh)^{1−d} ∫ e^{i(x′−y′)ξ′/h}a (x_{d}, x′, ξ′, h) u (x_{d}, y′) dy′dξ′. (3.12)

For A ∈ E_{cl}^{j} we shall note σ (a) = a_{0} the principal symbol of A et D^{j} ⊂ E the subspace of the differential operators.

Let Q̃_{1} = \frac{P_{φ}−P_{φ}^{∗}}{2i} = ImP_{φ} and Q̃_{2} = \frac{P_{φ}+P_{φ}^{∗}}{2} = ReP_{φ},with principal symbols 2q_{1} + 2φ′_{x_{d}}ξ_{d} = Imp_{φ} and q_{2} + 2ξ_{d}^{2} = Rep_{φ}. Consider, for f, g ∈ C_{0}^{∞} (K)

\begin{matrix}((f, g))_{0} = ∫_{ℝ_{+}^{d}} f\overline{g}dx, (f, g)_{0} = ∫_{ℝ^{d−1}} f |_{x_{d}=0} \overline{g} |_{x_{d}=0} dx′ \\ ((f, g))_{1} = ∫_{ℝ_{+}^{d}} h∇f\overline{∇g} + f\overline{g}dx, \\ (f, g)_{1} = ∫_{ℝ^{d−1}} h∇f |_{x_{d}=0} \overline{∇g} |_{x_{d}=0} +f |_{x_{d}=0} \overline{g} |_{x_{d}=0} dx′ \end{matrix} (3.13)

and ∥f∥_{0}, |f|_{0}, ∥f∥_{1}, |f|_{1}, the associated norms.

\begin{aligned}∥P_{φ}f∥_{0}^{2} & = ∥Q̃_{2}f∥_{0}^{2} + ∥Q̃_{1}f∥_{0}^{2} + i((Q̃_{1}f, Q̃_{2}f)) − i((Q̃_{2}f, Q̃_{1}f))_{0} \\ & = ∥Q̃_{2}f∥_{0}^{2} + ∥Q̃_{1}f∥_{0}^{2} + i(( [Q̃_{2}, Q̃_{1}] f, f)) + hβ (f) , \end{aligned} (3.14)

with (D_{x} = \frac{h∂_{x}}{i})

\begin{aligned}β (f) & = (Q̃_{1}f, D_{x_{d}}f)_{0} + ((D_{x_{d}} Q̃_{1} − 2φ′_{x_{d}} Q̃_{2}f) , f)_{0} \\ & = (2φ′_{x_{d}}D_{x_{d}}f, D_{x_{d}}f)_{0} + (A_{1}f, D_{x_{d}}f)_{0} + (A′_{1}D_{x_{d}}f, f)_{0} + (A_{2}f, f)_{0} \end{aligned} (3.15)

where A_{i} belong to D^{i}.

The principal symbol of the operator i [Q̃_{2}, Q̃_{1}] is h{Rep_{φ}, Imp_{φ}} and Gårding inequality gives from (3.4) (see [LR95b], paragraphe 3, lemme 2) that

∥P_{φ}f∥_{0}^{2} − Rehβ (f) ≥ ch (∥f∥_{0}^{2} + ∥h∇f∥_{0}^{2}) + h^{2}Re (D_{x_{d}}f + G_{1}f, G_{0}f)_{0} , (3.16)

with G_{0} ∈ E^{0} and G_{1} ∈ D^{1}, which implies 3.1. Noticing that with the hypotheses of proposition 3.2, near Γ_{D}, the only term appearing in (3.15) is (2φ′_{x_{d}}D_{x_{d}}f, D_{x_{d}}f)_{0} ≥ 0, we also obtain proposition 3.2 near Γ_{D}.

We now restrict ourselves to the case Γ_{N}. Note

E^{+} = \{(x, ξ′) ∈ K × ℝ^{d}; q_{2} (x, ξ′) + \frac{q_{1} (x, ξ′)^{2}}{(\frac{∂φ}{∂n})2} > 0\} (3.17)

Z = \{(x, ξ′) ∈ K × ℝ^{d}; q_{2} (x, ξ′) + \frac{q_{1} (x, ξ′)^{2}}{(\frac{∂φ}{∂n})2} = 0\} (3.18)

E^{−} = \{(x, ξ′) ∈ K × ℝ^{d}; q_{2} (x, ξ′) + \frac{q_{1} (x, ξ′)^{2}}{(\frac{∂φ}{∂n})2} > 0\} (3.19)

Following [LR95b], we shall use the technique of Calderón projectors (and elliptic calculus) in E^{+} and Carleman inequalities in Z and E^{−}.

The following result is proved in [LR95b] (lemme 5.3).

Lemma 3.4. — For (x, ξ′) ∈ E^{+}, p_{φ} has two roots, r_{±} with Imr_{+} > 0 and Imr_{−} < 0; for (x, ξ′) ∈ Z, one of the two roots is real and for (x, ξ′) ∈ E^{−}, the imaginary parts of the two roots has the same sign as ∂_{d}φ.

The following two lemma (lemme 5.4 and 5.5 of [LR95b]) are the keys to the proof of proposition 3.2 near Γ_{N}.

Lemma 3.5 (elliptic estimates). — Let χ ∈ S^{0}, with compact support in x. Assume that there exists δ > 0 such that on the support of χ one has

q_{2} (x, ξ′) + \frac{q_{1} (x, ξ′)^{2}}{(\frac{∂φ}{∂x_{d}} )2} ≥ δ (1 + |ξ′|^{2}) (3.20)

and ∂_{d}φ > 0 on {x_{n} = 0}. Then there exists h_{1}, C > 0 such that ∀0 < h < h_{0} and all f ∈ C_{0}^{∞} (K)

∥P_{φ}f∥_{0}^{2} + h|(hD_{d} − k)Op (χ) f|_{1}^{2} + h^{2}∥f∥_{1}^{2} ≥ C (∥Op (χ) f∥_{1}^{2} + h|Op (χ) f |_{x_{d}=0} |_{1}^{2}) . (3.21)

Lemma 3.6 (Carleman estimates). — Let φ satisfy conditions (3.2) and (3.3). Let χ ∈ S^{0}, with compact support in x. Assume that on the support of χ, |ξ′| is bounded and

(∂_{n}φ)^{2} − q_{2} > 0 (3.22)

Then there exists h_{1}, C > 0 such that ∀0 < h < h_{0} and all f ∈ C_{0}^{∞} (K)

(3.23) ∥P_{φ}f∥_{0}^{2} + h|(hD_{d} − k)Op (χ) f |_{x_{d}=0} |_{0}^{2} + h^{2}∥f∥_{1}^{2}

+ h|f |_{x_{d}=0} |_{0}|(hD_{d} − k)Op (χ) f |_{x_{d}=0} |_{0} + h^{2}|f |_{x_{d}=0} |_{0}^{2} ≥

C (h∥Op (χ) f∥_{1}^{2} + h|Op (χ) f |_{x_{d}=0} |_{1}) .

Now proposition 3.2 near Γ_{N} is a simple consequence of these two lemma. Indeed, it is enough to construct χ_{1} satisfying the hypotheses of lemma 3.5 and χ_{2} satisfying the hypotheses of lemma 3.6 such that χ_{1} + χ_{2} ≥ 1 and to sum the inequalities obtained (it is easy to see that the error terms can be absorbed). For |ξ′| large enough, (3.20) is satisfied and so the construction of χ_{1,2} is straightforward.

Lets give a sketch of the proof of lemma 3.5 and 3.6:

Set u = Op (χ_{1}) f, f_{1} = P_{φ}f and f_{0} = hD_{d}f − kf solutions of

P_{φ}u = Op (χ_{1}) f_{1} + [P_{φ}, Op (χ_{1})] f = u_{1}, |u_{1}| ≤ C (|f_{1}| + h∥f∥_{1})
(hD_{d}u − Op (k) u) |_{x_{d}=0}= u_{0}, |u_{0}|_{1} ≤ C (|f_{0}|_{1} + h|f |_{x_{d}=0} |_{1})
u_{0} = Op (χ_{1}) f_{0} + [hD_{d} − Op (k) , Op (χ_{1})] f |_{x_{d}=0}

(3.24)

Let \underline{u} be the 0 extension of u in x_{d} < 0 and γ_{0} (u) = u |_{x_{d}=0}, γ_{1} (u) = −hi∂_{n}u |_{x_{d}=0}. Taking into account that r_{+} + r_{−} = −2iφ′_{d}, we obtain by the jump formula,

P_{φ}\underline{u} = \underline{f}_{1} − h^{2}γ_{0} (u) ⊗ δ′_{x_{d}=0} + \frac{h}{i} [γ_{1} (u) − Op (r_{+} + r_{−}) γ_{0} (u)] ⊗ δ_{x_{d}=0} (3.25)

Using a parametrix of the operator P_{φ} (since u = Op(χ)f is supported in the elliptic region) we can solve equation (3.24) and taking again the traces on x_{d} = 0, we obtain by standard elliptic semi-classical calculous ( [LR95b], equation (3.48)): γ_{0} (u) = Op (a) γ_{0} (u) + Op (b) γ_{1} (u) + v (3.26) with a ∈ S^{0}, σ (Op (a)) = \frac{χ̃_{1}r_{−}}{r_{+}+r_{−}} |_{x_{d}=0}, χ̃_{1} equal to one near supp (χ_{1}), b ∈ S^{−1}, σ (Op (b)) = \frac{χ̃_{1}}{r_{+}+r_{−}} |_{x_{d}=0} and v satisfies |v|_{1} ≤ Ch^{−1/2} [∥f_{1}∥_{0} + h∥f∥_{1} + h^{2}|hD_{d}f |_{x_{d}=0} |_{0}] (3.27) Now γ_{0} (u) satisfies the equation [Op (1 − a) − Op (b) Op (k)] γ_{0} (u) = v + u_{0} (3.28) with (3.29) σ (Op (1 − a) − Op (b) Op (k)) |_{supp(χ_{1})}= \frac{r_{+} − k}{r_{+} + r_{−}} but, according to lemma 3.4, we have r_{+} − k = r_{+} + i∂_{d}φ ≠ 0, (3.30) so the operator (Op (1 − a) − Op (b) Op (k)) is elliptic, which gives the estimation |γ_{0} (u) |_{1} ≤ C [|u_{0}|_{1} + |v|_{1} + O (h^{∞}∥f∥_{1})] (3.31) and also (since γ_{1}(u) = u_{0} + kγ_{0}(u)) |γ_{1} (u) |_{1} ≤ C [|u_{0}|_{1} + |v|_{1} + O (h^{∞}∥f∥_{1})] (3.32) Using again the parametrix of the operator P_{φ}, equations (3.24), (3.29) and (3.32) we obtain the estimate (3.21)).

To prove lemma 3.6, we apply estimate (3.16) to the function u = Op (χ_{2}) f, satisfying (with f_{1} = P_{φ}f and f_{0} = hD_{d}f − Op (k) f)

P_{φ}u = Op (χ_{2}) f_{1} + [P_{φ}, Op (χ_{2})] f = u_{2}, |u_{2}| ≤ C (|f_{1}| + h∥f∥_{1})

(hD_{d}u − Op (k) u) |_{x_{d}=0}= u_{3}, |u_{3}|_{0} ≤ C (|f_{0}|_{0} + h∥f_{x_{d}=0}∥_{0})
u_{3} = Op (χ_{2}) f_{0} + [hD_{d} − Op (k) , Op (χ_{2})] f |_{x_{d}=0}

(3.33)

Now, β (u) = (Bu, u) + R (f, f_{0}) (3.34) with

\begin{aligned}σ (B) & = 2∂_{n}φ|k|^{2} − 2∂_{n}φq_{2} \\ & = 2∂φ_{n}(|∂_{n}φ|^{2} − q_{2}) \end{aligned} (3.35)

and

|R (f, f_{0}) | ≤ C [|f |_{x_{d}=0} |_{0}|f_{0}|_{0} + |f_{0}|_{0} + h|f |_{x_{d}=0} |_{0}^{2}] (3.36)

Since u = Op (χ_{2}) f and on the support of χ_{2}, one has σ (B) ≥ c > 0, by Gårding inequality, we obtain

(Bu, u) ≥ \frac{c}{2}|Op (χ_{2}) f |_{x_{d}=0} |_{1}^{2} (3.37)

and we get estimate(3.21)

Remark 3.7. — If the function g is supported in the exterior of a (fixed) neighbourhood of some part of the boundary of Ω, Γ_{0}, then proposition 3.2 is still true even if we no longer suppose that \frac{∂φ}{∂n} ≠ 0 on Γ_{0} since the proof is local and the result is empty (hence true) near Γ_{0}.

Remark 3.8. — Suppose that the phase φ can be choosen independently with respect to the parameter e ∈ [1, 2]. Then it is a consequence of the proof we present that the constants h_{1} and C can be choosen independantly with respect to e ∈ [1, 2] (in fact, from the proofs, these constants can be chosen locally independantly and a compacity argument gives the global independance).

4. Construction of phase functions

This section is devoted to the construction of the phase functions we shall use later to conjugate the operator P. For |x| large, we work in the coordinate system of section 2.2. In the sequel, except if explicitaly precised otherwise, every L^{2} norm considered will be the L^{2} norm for which the operator Q is self adjoint.

4.1. Construction of phases in any bounded domain. — Fix R > 0. In this part we shall construct two phases φ_{1} et φ_{2} satisfying the hypoellipticity conditions of Hörmander (3.2) in Ω∩B(0, R), except on a finite number of balls such that on the one hand, where one phase does not satisfy the hypotheses, the other one does and is strictly bigger than the first one, and on the other hand, far from the l’obstacle, the functions φ_{1} et φ_{2} are equal and radial. This construction is essentially done in [Bur98], the only difference is that we supposed that the operator P − e was equal to h^{2}Δ − 1 outside a compact set, whereas here we suppose only that the difference between P and −h^{2}Δ tends to 0.

Take R > 0 such that Θ ⊂ B (0, R). The following result is proved in[Bur98], proposition 3.1.

Proposition 4.1. — For R > 0 such that Θ ⊂ B (0, R), there exists two functions ψ_{1,2} ∈ C^{∞} (\overline{Θ^{c} ∩ B (0, R)}) such that ∂_{n}ψ_{i} |_{∂Θ}> 0 (∂_{n} is the interior normal to Θ^{c}), with only non degenerate critical points and such that on C (0, R), the circle centered in O with radius R, ∇ψ_{i} ·x > 0 and where ∇ψ_{i} = 0 (a finite number of points), one has ∇ψ_{i+1} ≠ 0 and ψ_{i+1} > ψ_{i} (ψ_{3} = ψ_{1}). Finally the functions ψ_{i} are radial and equal near C (0, R).

Remark 4.2. — We deduce that there exists a finite number of points, x_{i,j} ∈ Θ^{c}; i = 1, 2; j = 1, ⋯, N_{i}, a constant ε > 0 such that B (x_{i,j}, 2ε) ⊂ Θ^{c} ∩ B (0, R), B (x_{1,j}, 2ε) ∩ B (x_{2,j}, 2ε) = ∅ and on B (x_{i,j}, 2ε),ψ_{i} < ψ_{i+1} (ψ_{3} = ψ_{1}).

Note Ω_{i} = Θ^{c} ∩ B(0, R) ∩ (∪_{j}B (x_{i,j}, ε))^{c}.

We are looking for φ_{i} satisfying (3.2) on Ω_{i}, of the form e^{βψ_{i}}, β >> 1. It is enough to check that for ψ ∈ C^{∞} (\overline{Ω}) such that |∇ψ| ≥ c > 0 then for β > 0 large enough, {Rep_{φ}, Imp_{φ}} > c′ > 0 if p_{φ} (x, ξ) = 0.

We have φ′ = βe^{βψ}ψ′, φ′′ = e^{βψ} (β^{2}ψ′ ^{t}ψ′ + βψ′′) and if p_{φ} = 0 then^{t}ξAξ+a (x)·ξ = tφAφ + e + a (x) · φ′ hence |ξ| ≤ Cβe^{βψ}. The semi-classical principal symbol of the operator P has the following form:

\begin{matrix}p(x, ξ) = ^{t}ξA(x)ξ + b(x, h) · ξ + c(x, h), \\ p_{φ}(x, ξ) = ^{t}ξA(x)ξ − ^{t}φ′A(x)φ′ + b(x, h) · ξ + c(x, h) + 2ib(x, h) · φ′ + 2iA(x)φ′ \end{matrix} (4.1)

with A = (a_{i,j}(x)) uniformly definite positive, b and c real. A direct computation gives if p_{φ} (x, ξ) = 0,

{Rep_{φ}, Imp_{φ}} = 4e^{3βψ} (β^{4} (^{t}ψ′Aψ′)^{2} + O (β^{3})) , (4.2)

which implies the result since ψ′ ≠ 0 et A >> cId, c > 0.

Remark 4.3. — In the computation above, the choice of β (hence the choice of function φ_{1;2}) is clearly independant of the choice of e ∈ [1, 2].

4.2. Stabilisation of the phases outside of a ball. — We are going to define now the phases on B (0, R)^{c} ∩ B (0, R_{1}) (R_{1} to be fixed) so that near the set {|x| = R_{1}} the phases are small. This part is essentially the same as in [Bur98] except for minor technical modifications to take into account that the perturbation of the metric and the potential are not compactly supported.

At the level of the symbols, according to the reduction in section 2.2, if R is taken large enough, we can suppose that the operator P is as in (2.12) for |x| ≥ R, and that we have, for r ≥ R

\begin{matrix}|q_{2}(r, σ, \frac{η}{r})| ≤ \frac{1}{8} ∥\frac{η}{r} ∥^{2} ,  |r∂_{r}q_{2}(r, σ, \frac{η}{r})| ≤ \frac{1}{8} ∥\frac{η}{r} ∥^{2} \\ |q_{1}(r, σ, \frac{η}{r} , h)| ≤ \frac{1}{8} ∥\frac{η}{r} ∥ ,  |r∂_{r}q_{1}(r, σ, \frac{η}{r} , h)| ≤ \frac{1}{8} ∥\frac{η}{r} ∥ \\ |V (r, σ, h)| ≤ \frac{1}{8},  |r∂_{r}V | ≤ \frac{1}{8} \end{matrix} (4.3)

At the level of the operators, we shall also make the following assumptions on the choice of R: we suppose that for any r ≥ R and any function w on the sphere 𝕊^{d−1}

(4.4) |∫_{𝕊^{d−1}} 2q_{2}(r, σ, \frac{h∂_{σ}}{r})w\overline{w}| + |∫_{𝕊^{d−1}} q_{1}(r, σ, \frac{h∂_{σ}}{r})w\overline{w}|

+ |∫_{𝕊^{d−1}} r∂_{r}q_{2}(r, σ, \frac{h∂_{σ}}{r})w\overline{w}| + |∫_{𝕊^{d−1}} r∂_{r}q_{1}(r, σ, \frac{h∂_{σ}}{r})w\overline{w}|

≤ \frac{1}{8} (∫_{𝕊^{d−1}} |\frac{h∇_{σ}}{r} w|^{2} + |w|^{2})

and for any r ≥ R, z = re^{iθ}, θ ≤ θ_{0},

\begin{aligned}∫_{𝕊^{d−1}} |(q_{1}(z, σ, \frac{h∂_{σ}}{r} , h) − q_{1}(r, σ, \frac{h∂_{σ}}{r} , h)) w|^{2} & ≤ \frac{θ}{16} (∫_{𝕊^{d−1}} |\frac{h∇_{σ}}{r} w|^{2} + |w|^{2}) \\ ∫_{𝕊^{d−1}} |(q_{2}(z, σ, \frac{h∂_{σ}}{r} , h) − q_{2}(r, σ, \frac{h∂_{σ}}{r} , h)) w|^{2} & ≤ \frac{θ}{16} (∫_{𝕊^{d−1}} |\frac{h∇_{σ}}{r} w|^{2} + |w|^{2}) \end{aligned} (4.5)

Remark 4.4. — Of course, (4.4) is fulfilled if R is large enough because the coefficients of q_{2} and their derivatives with respect to σ are bounded by M/r^{β}, whereas the derivatives with respect to r are bounded by M/r^{β+1}; and the same remark holds for q_{1}. To check (4.5), remark that if q_{2}(r, σ, \frac{h∂_{σ}}{r}) = ∑ q_{2}^{i,j}(r, σ)h∂_{σ_{i}}h∂_{σ_{j}}, then

∑ q_{2}^{i,j}(z, σ) − ∑ q_{2}^{i,j}(r, σ) = (z − r) ∫_{0}^{1} ∂_{r}q_{2}^{i,j}(z + s(z − r), σ)ds (4.6)

and from the bound on the r-derivatives, we deduce

|∂_{2}^{α}∂_{1}q_{2}^{i,j}(z + s(z − r), σ)| ≤ \frac{M_{α}}{r^{β+1}} (4.7)

which implies

∫_{𝕊^{d−1}} |(q_{2}(z, σ, \frac{h∂_{σ}}{r} , h) − q_{2}(r, σ, \frac{h∂_{σ}}{r} , h)) w|^{2} ≤ \frac{M|z − r|}{|r|^{β+1}} (∫_{𝕊^{d−1}} |\frac{h∇_{σ}}{r} w|^{2} + |w|^{2}) (4.8)

The same argument gives also the result for the q_{1} term.

Since near the set {|x| = R} the phases φ_{1} and φ_{2} are equal, we will take the same extension for the two phases and dropp the index i. We are looking for radial extensions for r > R. The function p (r, θ, ρ, η) has the form:

p (r, σ, ϱ, η) = ϱ^{2} + \frac{∥η∥^{2}}{r^{2}} + q_{2} (r, σ, \frac{η}{r} ) + q_{1} (r, σ, \frac{η}{r} , h) + V (r, σ, h) − e (4.9)

and

p_{φ} (r, σ, ϱ, η) = p (r, σ, ϱ, η) − (φ′ (r))^{2} + 2iϱφ′(r). (4.10)

If p_{φ} = 0 (and φ′ (r) ≠ 0), we have

ϱ = 0 and \frac{∥η∥^{2}}{r^{2}} + q_{2} (r, σ, \frac{η}{r} ) + q_{1} (r, σ, \frac{η}{r} ) + V (r, σ, h) = (φ′ (r))^{2} + e (4.11)

which gives from (4.3)

(\frac{4}{5}(φ′)^{2} + \frac{1}{2}) ≤ ∥\frac{η}{r} ∥^{2} ≤ (\frac{4}{3}(φ′)^{2} + 3) (4.12)

and on the set {p_{φ} = 0}, we have, using that q_{2} is a quadratic form and q_{1} a linear form,

\begin{aligned}\{Rep_{φ}, Imp_{φ}\} & = 4(φ′)^{2}φ′′ + 2 (−∂_{1}q_{1} (r, σ, \frac{η}{r} ) − ∂_{1}q_{2} (r, σ, \frac{η}{r} ) − ∂_{r}V ) φ′ \\ & + \frac{φ′}{r} (4\frac{∥η∥^{2}}{r^{2}} + 4q_{2} (r, σ, \frac{η}{r} ) + 2q_{1} (r, σ, \frac{η}{r} )) \\ & = \frac{φ′}{r} (4\frac{∥η∥^{2}}{r^{2}} + 4q_{2} + 2q_{1} − 2r∂_{1}q_{2} − 2r∂_{1}q_{1} − 2∂_{r}V ) \\ & = \frac{φ′}{r} (4(\frac{∥η∥^{2}}{r^{2}} + q_{2} + q_{1} + V + rφ′φ′′)) \\ & − 2\frac{φ′}{r} (r∂_{1}q_{2} + r∂_{1}q_{1} + r∂_{r}V + 2V + q_{1}) \end{aligned} (4.13)

From (4.3)and (4.12) (4.14) |r∂_{1}q_{2} + r∂_{1}q_{1} + r∂_{r}V + 2V + q_{1}|

≤ (\frac{1}{8}∥η/r∥^{2} + \frac{1}{4}∥η/r∥ + \frac{3}{8}) ≤ (\frac{1}{4}∥η/r∥^{2} + \frac{1}{2})) ≤ \frac{(φ′)^{2}}{3} + \frac{5}{4}

and from (4.11) and (4.13) we obtain (on the set p_{φ} = 0)

(4.15) \{Rep_{φ}, Imp_{φ}\} ≥ \frac{φ′}{r} (4(φ′)^{2} + 4e − \frac{1}{3}(φ′)^{2} − \frac{5}{4} + 4rφ′φ′′)

≥ φ′ ( \frac{\frac{11}{3}(φ′)^{2} + \frac{11}{4}}{r} + 4φ′φ′′)

Our problem is reduced to the obtention of a phase function defined for r ≥ R and satisfying φ′ > 0 and

4φ′φ′′ + \frac{11}{4r} ≥ 0, φ′ (R) given (4.16)

In fact, we’ll construct φ such that

20φ′φ′′ + \frac{1}{r} ≥ 0, (4.17)

which implies (4.16).

The solution, k of the equation k′k + \frac{1}{24r} = 0 with k(R) = φ′(R) is k = \sqrt{(φ′ (R))^{2} − log (\frac{r}{R}) /12}. (4.18)

Take R_{1} = Re^{11φ′^{2}(R)}, κ > 0 small and define φ′ on Ω by

φ′ (r) = \begin{cases}φ′ (r) & if r < R \\ k(r) & if R ≤ r < R_{1}e^{−12κ^{2}} \\ κ & if R_{1}e^{−12κ^{2}}. \end{cases} (4.19)

κ  R  R_{1}e^{−κ^{2}/2}  R_{1}  R_{2}

Figure 1: The function φ′

Remark 4.5. — Remark that the constant R_{1} does depend only on the value of φ′(R), i.e. only on the geometry of the problem for |x| ≤ R (i.e. on the nature of the perturbation of the Laplace operator in a fixed compact set).

Let H ∈ C^{∞} (ℝ) equal to 0 for r < R − α/2 to 1 for r > R − α/3, 0 ≤ H ≤ 1, H′ ≥ 0 and h ∈ C_{0}^{∞} (]0, 1[), positive, of integral equal to 1. Note

h_{η} (r) = \frac{1}{η}h (\frac{r}{η} ) (4.20)

and

φ′_{η} = (1 − H) φ′ + h_{η} ∗ (Hφ′) (4.21)

The function φ′_{η} (r) converges uniformly to φ′ (r) (η → 0^{+} and

\frac{d}{dr} (φ′_{η}) (r) = (−H′) φ′ + h_{η} ∗ (H′φ′) + (1 − H) φ′′ + h_{η} ∗ (Hφ′′) (4.22)

but for r > R, φ′′ ≥ \frac{−1}{rφ′} and for R − α/3 ≤ r ≤ R, since the function φ has been built such that φ′′ >> 1, φ′′ ≥ \frac{−1}{rφ′}. We obtain (taking α > 0 small enough)

\frac{d}{dr} (φ′_{η}) (r) ≥ (−H′) φ′ + h_{η} ∗ (H′φ′) + (1 − H) \frac{−1}{rφ′} + h_{η} ∗ (H × \frac{−1}{rφ′} ) . (4.23)

The right hand side converges uniformly to \frac{−1}{rφ′}, hence there exists c > 0 such that if η > 0 is small enough, φ′′_{η}φ′_{η} + \frac{1}{22r} ≥ c > 0, for r ∈ [R − α, R_{2}].

Now, this construction allows us to construct phases φ_{1,2} satisfying the hypoellipticity assumptions of Hörmander (3.2), in Ω∩B (0, R_{2}), with φ_{1,2} radial and φ′ = κ arbitrarily small (independently of R_{2}) near {|x| = R_{2}}.

4.3. Construction of the phases near infinity. — In the preceding sections we have constructed phases satisfying the hypoellipticity assumptions of Hörmander and such that φ′ = κ > 0 for r ≥ R_{1}, the constant R_{1} being independant of the choice of κ > 0. In this section we arrive at the most delicate point of the construction of our phases: we are going to construct a family (depending on the parameter h > 0) of phases in {|x| ≥ R_{1}} equal to the phases constructed above near {|x| = R_{1}} and such that φ′ (r, h) = 8h/r for r very large. The difficulty in this construction is that we want to get some positivity in the bracket [ReP_{φ}, ImP_{φ}]. Of course, contrarily to the previous constructions which depends only on the principal symbol of the operator P, this construction does depend on the subprincipal symbol. Consequently, the fact that this construction is performed in the coordinate system of section 2 such that the operator satisfies (2.12) is important.

The most natural candidate for this construction would have been to take φ′(r) = k1_{r≤R_{1}} where k is defined by (4.18). A first remark is that we have to work with smooth phase functions. Note R_{2} = R_{1} + 4 and consider the function Ψ′(r) = e^{1/(r−R_{2})}1_{r≤R_{2}} ∈ C^{∞}(ℝ). Since near R_{2}, Ψ′ and Ψ′′ are small, Ψ satisfies

22ΨΨ′ + \frac{1}{r} ≥ 0 (4.24)

for |r − R_{2}| ≤ ε, ε > 0. Now suppose that κ > 0 has been chosen smaller than e^{−1/ε}. Then by smoothing the continuous function equal to φ for r ≤ R_{2} + log(κ) and equal to Ψ for r ≥ R_{2} + log(κ) we get

Lemma 4.6. — There exists a function φ̂ equal to φ for r ≤ R_{2} − 2ε and such that – i) 21φ̂′φ̂′′ + \frac{1}{r} ≥ 0 for R ≤ R_{2}

– ii) φ̂′(r) = e^{1/(r−R_{2})}1_{r≤R_{2}} near r = R_{2} and for r ≥ R_{2}.

Remark 4.7. — Since for r ≥ R_{2}, the function φ̂ is constant (φ̂′ = 0), it does no more satisfies the hypoellipticity assumptions and we have to modify it for r > R_{2} (and also for r close to R_{2}).

Consider a non negative increasing function χ_{1} ∈ C^{∞} equal to 0 for r ≤ 1, to 1 for r ≥ 10 and such that 0 ≤ χ′_{1} ≤ 1/8. We will also need to suppose that near {r = 1}, we have

χ_{1}(r) = e^{− \frac{1}{r−1}} 1 |_{r>1} (4.25)

Note

H(r, h) = \frac{h}{h^{1/2} + χ_{1}(r)} (4.26)

A direct calculus gives

Lemma 4.8. — There exists C > 0 such that ∀r ∈ ℝ, ∀0 < h < 1,

|H(r, h)| ≤ Ch^{1/2},  |H′(r, h)| ≤ C,  |H′′(r, h)| ≤ Ch^{−1/2}, (4.27)

Take a non negative function χ_{2} ∈ C_{0}^{∞}(] − 3, 12[), equal to 1 on [−1, 10] such that

|χ′_{2}| ≤ 1, |χ′′_{2}| ≤ 2 (4.28)

Lemma 4.9. — There exists a function g ≥ 0 such that – i) g is equal to 0 for r ≤ R_{2} + 1

– ii) g′ > 0 for r > R_{2} + 1

– iii) g′′ ≥ 2 and g′ ≥ 2 for r ∈ [R_{2} + 10, R_{2} + 12]

– iv) g′′ ≥ 0 for R_{2} + 1 ≤ r ≤ R_{2} + 12

– v) ∃R_{3} > R_{2}; ∀r ≥ R_{3}, 8/r = g′(r)

– vi) ∀r ≥ R_{2} + 12, (g′)^{2} + 7hg′′ ≥ 0 and g′ ≥ \frac{7}{r}

Indeed, the only serious restrictions on g are imposed by conditions v) and vi), but the solution to (k′)^{2} + 7k′′ = 0 is k = 7/r and (8/r)^{2} + 7(8/r)′ = 8/r^{2} > 0 which shows that condition v) is compatible with vi). From now on, we shall note φ the function equal to φ̂ for r ≤ R_{1} and such that

φ′(r) = φ̂′(r) + χ_{2}(r − R_{2})H(r − R_{2}, h) + hg′(r) (4.29)

Lemma 4.10. — For any ε > 0 there exists h_{0} > 0 such that for any 0 < h < h_{0} and any r ≥ R_{1}

h|φ′′′| ≤ ε\frac{φ′}{r} , h|φ′′|^{2} ≤ ε\frac{φ′}{r} (4.30)

The proof of this lemma depends on the position of the point r:

– i) For r ≥ R_{3} φ′(r) = \frac{8h}{r} and the result follows by a simple computation

– ii) For R_{2} + 10 ≤ r ≤ R_{3}, according to lemma 4.9 iii) and vi), we have

φ′ = Hχ_{2} + hg′ ≥ hg′ ≥ h inf(2, \frac{7}{R_{3}} ) (4.31)

|φ′′| + |φ′′′| ≤ Ch and the result follows by compacity.

– iii) For R_{2} + 1 ≤ r ≤ R_{2} + 10,

\begin{aligned}φ′ & = \frac{h}{h^{1/2} + χ_{1}} + hg′ \\ φ′′ & = −\frac{h}{(h^{1/2} + χ_{1})^{2}} χ′_{1} + hg′′ \\ φ′′′ & = 2\frac{h}{(h^{1/2} + χ_{1})^{3}} (χ′_{1})^{2} − \frac{h}{(h^{1/2} + χ_{1})^{2}} χ′′_{1} + hg′′′ \end{aligned} (4.32)

Close to r = R_{2} + 1 we have

χ_{1} = e^{− \frac{1}{r−(R_{2}+1)}} 1 |_{r≥R_{2}+1} (4.33)

Choose 3 < α < 4.

1. For R_{2} + 1 − \frac{α}{log h} ≤ r, we have χ_{1} ≥ h^{1/α}. From (4.32) we obtain since

α > 3,

h|φ′′′| ≤ Ch^{2−3/α} + Ch^{2−2/α} + Ch^{2} = o(h) (4.34)

Since φ′ ≥ ch (with c > 0) we obtain the first inequality of lemma 4.10 (for R_{2} + 1 − \frac{α}{log h} ≤ r). The second inequality is obtained similarly.

2. For R_{2} + 1 ≤ r ≤ R_{2} + 1 − \frac{α}{log h}. Choose η > 1 such that αη < 4. There exists ε > 0 and C > 0 such that for 0 < x < ε we have

\frac{1}{x^{2}} e^{− \frac{1}{x}} ≤ Ce^{− \frac{1}{ηx}} ,  \frac{1}{x^{3}} e^{− \frac{1}{x}} ≤ Ce^{− \frac{1}{ηx}} ,  \frac{1}{x^{4}} e^{− \frac{1}{x}} ≤ Ce^{− \frac{1}{ηx}} (4.35)

This gives for R_{2} + 1 ≤ r ≤ R_{2} + 1 − \frac{α}{log h}, according to (4.33),

\begin{aligned}|χ′_{1}| & ≤ Ce^{\frac{log h}{ηα}} = h^{\frac{1}{ηα}} \\ |χ′′_{1}| & ≤ Ce^{\frac{log h}{ηα}} = h^{\frac{1}{ηα}} \end{aligned} (4.36)

From (4.32) we obtain since αη < 4,

h|φ′′′| ≤ Ch^{2−3/2+2/(ηα)} + Ch^{2−1+1/(ηα)} + Ch^{2} = o(h) (4.37)

and we obtain the first inequality. The second one is obtained similarly.

– iv) For R_{2} − 1 ≤ r ≤ R_{2} + 1,

φ′ = φ̂′ + h^{1/2} ≥ h^{1/2}  φ′′ = φ̂′′  φ′′′ = φ̂′′′ (4.38)

and the result is obvious

– v) For ≤ r ≤ R_{2} − 1, φ′ = φ̂′ + χ_{2}h^{1/2} ≥ c > 0, |φ′′| ≤ C, |φ′′′| ≤ C.

Proposition 4.11. — There exists h_{0} > 0 such that for any 0 < h < h_{0} and any r ≥ R_{1}, we have

φ′(\frac{1}{r} + 20φ′φ′′) ≥ 0 (4.39)

According to lemma 4.10 it is enough to prove

φ′(\frac{1}{r} + \frac{41}{2}φ′φ′′) ≥ 0 (4.40)

The proof of this result depends again on the position of the point r:

– i) For r ≥ R_{3}, φ′ = hg′ = \frac{8h}{r} and a simple computation gives the result.

– ii) For R_{2} ≤ r ≤ R_{3}, we have

|φ′| ≤ Ch^{1/2}, |φ′′| ≤ C, (4.41)

and the result follows

– iii) For R_{2} − 3 ≤ r ≤ R_{2}, φ′ = φ̂′ + h^{1/2}χ_{2}(r − R_{2}) which implies

\begin{matrix}φ′ = φ̂′ + O(h^{1/2}) \\ φ′′ = φ̂′′ + O(h^{1/2}) \end{matrix} (4.42)

and we obtain from lemma 4.6, i),

φ′(\frac{1}{r} + \frac{41}{2}φ′φ′′) ≥ φ′(\frac{1}{40r} − Ch^{1/2}) (4.43)

which, by a compacity argument implies proposition (4.40).

– iv) For r ≤ R_{2} − 3, φ = φ̂′ and the result follows from lemma 4.6 i).

Proposition 4.12. — There exists h_{0} > 0 such that for any 0 < h < h_{0}, we have

(φ′)^{2} − 2h\frac{φ′}{r} + 4hφ′′ − h^{2}|φ′′′| ≥ 0 (4.44)

The proof of this proposition depends on the position of r.

– i) For R_{3} ≤ r, φ′ = hg′ = \frac{8h}{r} and a simple computation gives the result.

– ii)For R_{2} + 12 ≤ r ≤ R_{3}, φ′ = hg′ and from lemma 4.9 we have

\frac{h}{r} ≤ \frac{φ′}{7} , , 4φ′′ ≥ \frac{−4}{7}(φ′)^{2} (4.45)

and from lemma 4.10 (with ε < 1), we obtain

(4.46) (φ′)^{2} − 2h\frac{φ′}{r} + 4hφ′′ − h^{2}|φ′′′| ≥ h^{2}(g′)^{2}(1 − \frac{2}{7} − \frac{4}{7}) − h^{3}|g′′′|

≥ h^{2} \frac{(g′)^{2}}{7} − h^{3}|g′′′| ≥ h^{2}(g′)^{2}(\frac{1}{7} − \frac{ε}{r})

– iii) For R_{2} + 10 ≤ r ≤ R_{2} + 12, φ′ = hg′ + hχ_{2} ≥ 2h from lemma 4.9, iii); and since from lemma 4.8, g′′ ≥ 2 and |χ′_{2}| ≤ 1, we have φ′′ ≥ 0 and

(φ′)^{2} − 2h\frac{φ′}{r} + 4hφ′′ − h^{2}|φ′′′| ≥ (φ′)^{2} − 2h\frac{φ′}{r} − Ch^{3} ≥ 2h (2h − \frac{2h}{10}) − Ch^{3}. (4.47)

– iv) For R_{2} + 1 ≤ r ≤ R_{2} + 10, φ′ = H + hg′ ≥ H ≥ h − Ch^{3/2} and since R_{2} ≥ 12,

φ′ − \frac{2h}{r} ≥ \frac{2}{3}φ′ (4.48)

Since g′ ≥ 0, g′′ ≥ 0

(φ′)^{2} − 2h\frac{φ′}{r} + 4hφ′′ − h^{2}|φ′′′| ≥ \frac{2}{3}H^{2} + 4hH′ − h^{2}|H′′| − Ch^{3} (4.49)

But

hH′ = −H^{2}χ′_{1}, h^{2}H′′ = 2H^{3}(χ′_{1})^{2} − hH^{2}χ′′_{1} (4.50)

and since |χ′_{1}| ≤ 1/8, we obtain

\begin{aligned}(φ′)^{2} − 2h\frac{φ′}{r} + 4hφ′′ − h^{2}|φ′′′| & ≥ H^{2}(\frac{2}{3} − \frac{1}{2} − H|χ′_{1}|^{2} − h|χ′′_{1}|) − Ch^{3} \\ & ≥ H^{2}(\frac{1}{6} − Ch^{1/2}) − Ch^{3} ≥ \frac{h^{2}}{6} − Ch^{5/2} \end{aligned} (4.51)

– v) For R_{2} ≤ r ≤ R_{2} + 1

φ′ = h^{1/2} + hg′ (4.52)

and

(φ′)^{2} − 2h\frac{φ′}{r} + 4hφ′′ − \frac{h^{2}}{2} |φ′′′| = h + O(h^{3/2}). (4.53)

– vi) For r ≤ R_{2}, φ′ = φ̂′ + χ_{2}h^{1/2}. Since φ̂ = 0 for r ≥ R_{2} there exists 1 > α > 0 such that for R_{2} − α ≤ r ≤ R_{2},

\frac{φ̂′}{r} ≤ \frac{1}{8} and |φ̂′′| ≤ \frac{1}{8} (4.54)

Then (for R_{2} − α ≤ r ≤ R_{2}), since χ_{2} = 1, φ′ ≥ h^{1/2},

(φ′)^{2} − 2h\frac{φ′}{r} + 4hφ′′ − \frac{h^{2}}{2} |φ′′′| ≥ h − \frac{h}{4} − \frac{h}{4} − Ch^{3/2} (4.55)

For r ≤ R_{2} − α, φ̂′ ≥ c > 0 and

(φ′)^{2} − 2h\frac{φ′}{r} + 4hφ′′ − \frac{h^{2}}{2} |φ′′′| ≥ c^{2} − Ch^{1/2}. (4.56)

Proposition 4.13. — There exists C > 0 and h_{0} > 0 such that for any 0 < h < h_{0} and any r ≥ R_{1},

|φ′′| ≤ C,  |φ′′| ≤ Ch^{−1/2}φ′,  |φ′′′| ≤ Ch^{−3/2}φ′ (4.57)

The proof of the first inequality is straightforward. For the others we distinguish three cases:

– i) For r ≤ R_{2}, we have φ′ ≥ h^{1/2} and

|φ′′| ≤ C, |φ′′′| ≤ C (4.58)

– ii) For R_{2} ≤ r ≤ R_{3}, we have φ′ ≥ H ≥ h − Ch^{3/2} and

|φ′′| ≤ C, |φ′′′| ≤ Ch^{−1/2} (4.59)

– iii) For r ≥ R_{3} we have φ′ = hg′ = \frac{8h}{r} and the result follows.

5. Localization

Consider the phase functions, φ_{1} and φ_{2} constructed in section 4. Perform the analytic dilation as in section 2.3 for the operator P, with B ≥ R_{3} (R_{3} is defined in lemma 4.9). Note A = R_{1} (R_{1} is defined in section 4.2). Consider the operator P_{θ_{0}} obtained. Take χ_{1} ∈ C_{0}^{∞}(ℝ^{d}) equal to 1 for |x| ≤ A and with support in {x; |x| ≤ A+1}. From the (global) Carleman estimates of proposition 3.2, we obtain Proposition 5.1. — There exists c, C > 0, h_{0} > 0, ∀0 < h < h_{0}, e ∈ [1, 2], ∀u ∈ (D_{θ_{0}})

(5.1) ∫_{Ω}(e^{2φ_{1}/h}+e^{2φ_{2}/h})|χ_{1}(P −e)u|^{2}drdL ≥ ch ∫_{Ω}(e^{2φ_{1}/h}+e^{2φ_{2}/h})(|χ_{1}u|^{2}+|h∇χu|^{2})drdL

− Ch^{2} ∫_{A≤|x|≤A+1} e^{2φ/h} (|h∇u|^{2} + |u|^{2}) drdL

Indeed, to prove proposition 5.1 remark that

\begin{aligned}_{∫Ω} & (e^{2φ_{1}/h} + e^{2φ_{2}/h})|χ_{1}(P − e)u|^{2} \\ & ≥ \frac{1}{2} ∫_{Ω}(e^{2φ_{1}/h} + e^{2φ_{2}/h})|(P − e)χ_{1}u|^{2} − 2 ∫_{Ω}(e^{2φ_{1}/h} + e^{2φ_{2}/h})|[P, χ_{1}]u|^{2} \\ & ≥ \frac{1}{2} ∫_{Ω}(e^{2φ_{1}/h} + e^{2φ_{2}/h})|(P − e)χ_{1}u|^{2} − Ch^{2} ∫_{A≤r≤A+1} e^{2φ/h}(|h∇u|^{2} + |u|^{2}) \end{aligned} (5.2)

Consider cut-off Ψ_{1}, Ψ_{2} supported respectively in Ω_{1} ∪{R ≤ |x| ≤ A+1} and Ω_{2} ∪{R ≤ |x| ≤ A + 1} (the sets Ω_{i} are defined in remark 4.2) and such that Ψ_{1}^{2} + Ψ_{2}^{2} = 1 on the support of χ_{1}. Apply proposition 3.2 to function Ψ_{i}χ_{1}u and phases φ_{i}. and φ_{2}. Remark that due to the (finite number of) critical points of the functions φ_{i}, some error terms due to cut-off appear. However, when one sums the two inequalities obtained on Ω_{1;2} ∪ {R ≤ |x| ≤ A + 1} with φ_{1} and φ_{2}, since on these small balls around the critical points of φ_{i}, φ_{i+1} has no critical point and φ_{i+1} > φ_{i} (φ_{3} = φ_{1}), the error terms due to cut-off Ψ_{i} to avoid these balls can be absorbed. See [Bur98, section 3.2] for a detailled analysis.

Consider now χ_{2} ∈ C_{0}^{∞}(Γ_{θ_{0}}) and χ_{3} ∈ C^{∞}(Γ_{θ_{0}}) such that 1 = χ_{1} + χ_{2} + χ_{3}, and

(5.3) χ_{2} = \begin{cases}0 for |x| ∉ [A, B + 1] \\ 1 for |x| ∈ [A + 1/2, B + 1/2] \end{cases} , χ_{3} = \begin{cases}0 for |x| ≤ B + 1/2 \\ 1 for |x| ≥ B + 1 \end{cases}

The rest of the article will be devoted to the proof of the following result (remark that on the support of χ_{2}, φ_{1} = φ_{2}). We recall that the L^{2} norms we consider are the natural norms for which the operator P is self adjoint. For |x| ≥ A, the corresponding integration measure is simply the measure drdL.

Proposition 5.2. — There exists c, C > 0, h_{0} > 0, ∀0 < h < h_{0}, e ∈ [1, 2], ∀u ∈ (D_{θ_{0}}), with v = e^{φ/h}χ_{2}u,

(5.4) ∫ e^{2φ/h}|χ_{2}(P_{θ} − e)u|^{2} − e^{2φ(B,h)/h}Im (e^{iκ_{θ_{0}}} (P_{θ} − e)u, u)_{L^{2}} ≥

c (∥(φ′)^{1/2}h∂_{r}v∥_{L^{2}}^{2} + ch∥(\frac{φ′}{r})^{1/2}h\frac{∇σ}{r} v∥_{L^{2}}^{2} + h∥(\frac{φ′}{r})^{1/2}v∥_{L^{2}}^{2})

+ ce^{2φ(B,h)/h} ∫ κ_{θ_{0}}(|h∇u|^{2} + |u|^{2}) − Ch^{2} ∫_{A≤|x|≤A+1} e^{2φ/h}|h∇u|^{2} + |u|^{2}

Remark 5.3. — The last but one term in the estimate (5.4) is an elliptic estimate and corresponds to the fact that for r > B (e.g. after the analytic dilation) the operator P_{θ_{0}} − e is elliptic (in the semi-classical meaning). Taking into account that for A ≤ r ≤ B the phase φ′ varies from φ′ ≥ c > 0 for r near A to φ′ > Ch for r near B. The first terms in the right hand side of (5.4) correspond to subelliptic estimate with loose of 1/2 derivatives near A (Carleman type estimate) and 1 derivative near B (Mourre type estimate). Finally the last term is an error term due to cut-off.

Admitting this proposition for a while, we are in position to prove theorem 3. Consider u, f ∈ L^{2}(Γ_{θ_{0}}) solutions to

(P_{θ} − e)u = f (5.5)

Note φ̃ a function equal to max(φ_{1}, φ_{2}) for r ≤ B + 2, constant for r ≥ B + 3 and such that for r ≥ B,

|φ(r, h) − φ_{1;2}(B, h))| ≤ Ch (5.6)

Summing (5.1) and (5.4) we get

\begin{aligned}4 & ∫_{Γ_{θ}} e^{2φ̃/h}|(P_{θ_{0}} − e)u|^{2} − e^{2φ(B,h)/h}Im (e^{iκ_{θ_{0}}} (P_{θ_{0}} − e)u, u)_{L^{2}(Γ_{θ_{0}})} \\ & ≥ ch ∫_{Γ_{θ_{0}}} e^{2φ̃/h}(|χ_{1}u|^{2} + |h∇χ_{1}u|^{2}) \\ & + c (∥(φ̃′)^{1/2}e^{φ̃/h}h∂_{r}χ_{2}u∥_{L^{2}}^{2} + h∥(\frac{φ̃′}{r})^{1/2}e^{φ̃/h}h\frac{∇σ}{r} χ_{2}u∥_{L^{2}}^{2} + h∥(\frac{φ̃′}{r})^{1/2}e^{φ̃/h}χ_{2}u∥_{L^{2}}^{2}) \\ & + ce^{2φ(B,h)/h} ∫ κ_{θ_{0}}(|h∇u|^{2} + |u|^{2}) − Ch^{2} ∫_{A≤r≤A+1} e^{2φ̃/h}(|h∇u|^{2} + |u|^{2}) \end{aligned} (5.7)

But taking into account that for A ≤ r ≤ A + 1, we have φ′ ≥ α > 0, we easilly see that the last term can be absorbed by the four first ones (provided that h > 0 is small enough) and we obtain

\begin{aligned}4 & ∫_{Γ_{θ}} e^{2φ̃/h}|(P_{θ_{0}} − e)u|^{2} − e^{2φ̃(B,h)/h}Im (e^{iκ_{θ_{0}}} (P_{θ_{0}} − e)u, u)_{L^{2}} \\ & ≥ ch ∫ e^{2φ̃/h}(|χ_{1}u|^{2} + |h∇χ_{1}u|^{2}) + ce^{2φ̃(B,h)/h} ∫ κ_{θ_{0}}(|h∇u|^{2} + |u|^{2}) \\ & + c (∥(φ̃′)^{1/2}e^{φ̃/h}h∂_{r}χ_{2}u∥_{L^{2}}^{2} + h∥(\frac{φ̃′}{r})^{1/2}e^{φ̃/h}h\frac{∇σ}{r} χ_{2}u∥_{L^{2}}^{2} + h∥(\frac{φ̃′}{r})^{1/2}e^{φ̃/h}χ_{2}u∥_{L^{2}}^{2}) \end{aligned} (5.8)

or

(5.9) ∥e^{φ̃/h}(P_{θ_{0}} − e)u∥_{L^{2}}^{2} + e^{−φ̃(B,h)/h}∥e^{−φ̃/h}(P_{θ_{0}} − e)∥_{L^{2}}∥e^{φ̃/h}u∥_{L^{2}}

≥ ch^{2}(∥e^{φ̃/h}u∥_{L^{2}}^{2} + ∥e^{φ̃/h}h∇u∥_{L^{2}}^{2})

which gives, since 0 ≤ φ̃ ≤ Cte,

∥e^{φ̃/h}u∥_{L^{2}}^{2} ≤ \frac{C′}{h^{2}} (e^{2φ̃(B,h)/h}∥e^{−φ̃/h}(P_{θ_{0}} − e)u∥_{L^{2}}^{2} + ∥e^{φ̃/h}(P_{θ_{0}} − e)u∥_{L^{2}}^{2}) (5.10)

and we obtain theorem 3.

Suppose now that (P_{θ_{0}} − e)u is supported in the set R_{2} + 12 ≤ r, where φ(r, h) = φ(B, h) + O(h). Then, since

|e^{2φ(B,h)/h}Im (e^{iκ_{θ_{0}}} (P_{θ_{0}} − e)u, u)_{L^{2}(Γ_{θ_{0}})}| ≤ ∥e^{φ̃/h}(P_{θ_{0}} − e)u∥_{L^{2}}∥e^{φ̃/h}u∥_{L^{2}} (5.11)

from (5.10) we obtain

∥e^{φ̃/h}u∥_{L^{2}}^{2} ≤ \frac{C′}{h^{2}} ∥e^{φ̃/h}(P_{θ_{0}} − e)u∥_{L^{2}}^{2} (5.12)

and we obtain theorem 4: indeed, since for R_{2} + 12 ≤ r ≤ B, we have φ(r, h) = φ(B) + O(h), (5.12) implies the following

∥1_{R_{2}+12≤r≤B}(P_{θ_{0}} − e)^{−1}1_{R_{2}+12≤r≤B}f∥_{L^{2}}^{2} ≤ \frac{C}{h^{2}} ∥f∥_{L^{2}}^{2} (5.13)

Since

1_{R_{2}+12≤r≤B}(P_{θ_{0}} − e)^{−1}1_{R_{2}+12≤r≤B} = 1_{R_{2}+12≤r≤B}(P − e)^{−1}1_{R_{2}+12≤r≤B} (5.14)

where (P −e)^{−1} is the outgoing resolvent, we obtain theorem 3 (for the ring {R_{2} +12 ≤ r ≤ B}); and to conclude it is enough to remark that in the constructions above, B can be taken arbitrarily large (in which case the constant C depends on B).

6. The case of a potential

In this section we give the proof of proposition 5.2 for the simpler case where the operator we consider has the following form:

Q = −h^{2}Δ + V (x, h) (6.1)

in the exterior of a compact set.

In this case the reductions of appendix A are straightforward: Take R + 1 ≤ R_{1} = A ≤ R_{3} ≤ B and suppose that R is large enough so that for r ≥ R the operator P is as in (6.1). Consider ϱ a smooth function equal to 1 for r ≤ A − 1 and to r^{(d−1)/2} for r ≥ A. The operator ϱP ϱ^{−1} has the following form:

(6.2) P = −h^{2}(∂_{r}^{2} + Δ_{σ}) + h (2ϱ^{−1}ϱ′ − \frac{d − 1}{r}) h∂_{r}

+ h^{2} (\frac{d − 1}{r}ϱ^{−1}ϱ′ − 2ϱ^{−2}(ϱ′)^{2} + ϱ^{−1}ϱ′′) u

hence for r ≥ A,

P = ϱQϱ^{−1} = −h^{2}(∂_{r}^{2} + \frac{Δ_{σ}}{r^{2}} ) + Ṽ (x, h) (6.3)

where

Ṽ (x, h) = V (x, h) + h^{2} \frac{(d − 1)(d − 3)}{4r^{2}} (6.4)

For the sake of concisness, we shall still note Ṽ, V (since for our purpose Ṽ satisfies exactly the same assumptions as V). With this notation the operator P is self adjoint for the measure μ = ϱ^{−2}r^{d−1}drdL where dL is the Lebesgue measure on the d − 1 sphere. We’ll note

(u, v)_{L^{2}} (6.5)

The L^{2} scalar product for this measure.

We fix θ_{0} and note P_{θ_{0}} the operator obtained from P by the analytic dilation process for r ≥ B as in section 2. Consider 2 functions u ∈ D_{θ_{0}} and f ∈ L_{θ_{0}}^{2} with supports in the set {A < r} and such that (for a number e ∈ [1, 2]),

(P_{θ_{0}} − e)u = f (6.6)

6.1. Dissipation estimates. —

Proposition 6.1. — For any δ > 0 there exists h_{0} such that for any 0 < h < h_{0} and u ∈ D_{θ_{0}}

(6.7) − Im (e^{iκ_{θ_{0}}(r)}(P_{θ_{0}} − e)u, u)_{L^{2}}

≥ \frac{1}{2} ∫ (κ_{θ_{0}}(r)+rκ_{θ_{0}}(r)′)|h∂_{r}u|^{2} +κ_{θ_{0}}(r)(|\frac{h∇_{σ}}{r} u|^{2} +|u|^{2})drdL−h^{2}ε(δ) ∫_{B≤r≤B+δ} |u|^{2}

with

lim_{δ→0} ε(δ) = 0 (6.8)

Take Ψ ∈ C^{∞}(ℝ^{d}) equal to 0 for |x| ≤ A and equal to 1 for |x| ≥ A + 1. Remark that since the operator P is self adjoint,

Im (e^{iκ_{θ_{0}}(r)}(P_{θ_{0}} − e)u, u)_{L^{2}}
= Im (e^{iκ_{θ_{0}}(r)}(P_{θ_{0}} − e)Ψu, Ψu)_{L^{2}} + Im ((P − e)(1 − Ψ)u, Ψu)_{L^{2}}
+ Im ((P − e)(1 − Ψ)u, (1 − Ψ)u)_{L^{2}} + Im ((P − e)Ψu, (1 − Ψ)u)_{L^{2}}
= Im (e^{iκ_{θ_{0}}(r)}(P_{θ_{0}} − e)Ψu, Ψu)_{L^{2}}

(6.9)

According to this remark, to compute Im (e^{iκ_{θ_{0}}(r)}(P_{θ_{0}} − e)u, u)_{L^{2}}, we can replace u by Ψu. In the following we still note u, but suppose that u is supported in the set {|x| ≥ A}. We have

P_{θ_{0}} = − (\frac{e^{−iκ_{θ_{0}}(r)}}{1 + irκ_{θ′_{0}}(r)}h∂r)^{2} − h^{2}e^{−2iκ_{θ_{0}}(r)} \frac{h^{2}Δ_{σ}}{r^{2}} + V (re^{iκ_{θ_{0}}(r)}, σ, h) (6.10)

and from (2.19),

(6.11) − Im (e^{iκ_{θ_{0}}(r)}(P_{θ_{0}} − e)u, u)_{L^{2}}

= ∫ Im (−\frac{e^{−iκ_{θ_{0}}(r)}}{(1 + irκ′_{θ_{0}}(r))^{2}} ) |h∂_{r}u|^{2} + Im (−e^{−iκ_{θ_{0}}(r)}) |h\frac{∇_{σ}}{r} u|^{2}drdL

+ ∫ Im (e^{iκ_{θ_{0}}(r)}(e − V (re^{iκ_{θ_{0}}(r)}, σ, h))) |u|^{2}drdL − hIm(∫ g(r)h∂_{r}u\overline{u}drdL)

≥ ∫ \frac{3}{4}(κ_{θ_{0}}(r)+rκ′_{θ_{0}}(r))|h∂_{r}|^{2}+κ_{θ_{0}}(r)(\frac{3}{4}|h\frac{∇_{σ}}{r} u|^{2}+\frac{1}{2}|u|^{2})drdL−hIm(g(r)h∂_{r}u\overline{u})drdL

with

g(r) = ∂_{r} (\frac{1}{1 + irκ′_{θ_{0}}(r)}) \frac{e^{−iκ_{θ_{0}}(r)}}{1 + irκ′_{θ_{0}}(r)}. (6.12)

We have used here that

V (re^{iκ_{θ_{0}}(r)}) = V (r) + r(e^{iκ_{θ_{0}}(r)} − 1)∂_{r}V (z) (6.13)

for some z ∈]r, re^{iκ_{θ_{0}}(r)}[ and since V (r) is real we obtain from (4.3),

|Im(e^{iκ_{θ_{0}}(r)}V (re^{iκ_{θ_{0}}(r)}))| ≤ |κ_{θ_{0}}(r)||V (r)| + |κ_{θ_{0}}(r)||r∂_{r}V (z)| ≤ \frac{|κ_{θ_{0}}(r)|}{4}. (6.14)

from which we deduce (using (2.19))

|Im(e^{iκ_{θ_{0}}(r)}(e − V (re^{iκ_{θ_{0}}(r)})))| ≥ \frac{1}{2}|κ_{θ_{0}}(r)|. (6.15)

An integration by parts shows that the last term in (6.11) is smaller (in absolute value) than

\frac{h}{2} ∫ |Reg||h∂_{r}u||u| + \frac{h^{2}}{2} ∫ |∂_{r}g||u|^{2} (6.16)

but

g(r) = \frac{−i(rκ′′_{θ_{0}}(r) + κ′_{θ_{0}}(r))e^{−iκ_{θ_{0}}(r)}}{(1 + irκ′_{θ_{0}}(r))^{2}} (6.17)

and from (2.18)

\begin{aligned}|Reg| & ≤ C(|κ′_{θ_{0}}(r)| + |κ′′_{θ_{0}}(r)|)(|κ_{θ_{0}}(r)| + |κ′_{θ_{0}}(r)|) ≤ C|κ_{θ_{0}}(r)|, \\ |∂_{r}g| & ≤ C (|κ′′′_{θ_{0}}(r)| + |κ′′_{θ_{0}}(r)| + |κ_{θ_{0}}(r)′|) \end{aligned} (6.18)

The contribution of the first term in (6.16) is smaller than

Ch ∫ κ_{θ_{0}}(r) (|h∂_{r}u|^{2} + |u|^{2}) . (6.19)

To study the second term we write

h^{2} ∫ |∂_{r}g||u|^{2} = h^{2} ∫_{B≤r≤B+δ} |∂_{r}g||u|^{2} + h^{2} ∫_{B+δ≤r} |∂_{r}g||u|^{2} (6.20)

The second term in the right hand side of (6.20) is smaller than

C(δ)h^{2} ∫ κ_{θ_{0}}(r)|u|^{2} (6.21)

and since for r ≥ δ, κ_{θ_{0}}(r) ≥ κ_{θ_{0}}(r)(δ) > 0, it can be (for fixed δ > 0) absorbed by the right hand side of (6.11); whereas the first term in the righthand side of (6.20) is smaller than

ε(δ)h^{2} ∫_{B≤r≤B+δ} |u|^{2}, with ε(δ) = sup_{B≤r≤B+δ} |∂_{r}g| (6.22)

and (from (6.18))

lim_{δ→0} ε(δ) = 0. (6.23)

From (6.11), (6.19), (6.20), (6.21) and (6.22), we obtain proposition 6.1.

6.2. Commutators estimates. — In this section we prove positive commutator type estimates for a conjugate of the operator P.

Consider the function φ(r, h) constructed in section 4. Note

\begin{aligned}P_{φ} & = e^{φ/h}P e^{−φ/h} \\ & = −h^{2}∂_{r}^{2} − \frac{h^{2}Δ_{σ}}{r^{2}} + V − (φ′)^{2} + i (φ′h\frac{∂_{r}}{i} + h\frac{∂_{r}}{i} φ′) \\ & = ReP_{φ} + iImP_{φ} \end{aligned} (6.24)

Remember that χ_{2} ∈ C_{0}^{∞}(ℝ^{d}) with support in {A ≤ r ≤ B + 1} and equal to 1 in {A + 1/2 ≤ r ≤ B + 1/2}.

Proposition 6.2. — There exists h_{0} > 0 such that for any 0 < h < h_{0}, e ∈ [1, 2] and any u ∈ D_{κ_{θ_{0}}(r)}, if v = χ_{2}ue^{φ/h}

2∥(P_{φ} − e)v∥_{L^{2}}^{2} ≥ ∥φ′h∂_{r}v∥_{L^{2}}^{2} + h (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} + \frac{h}{2} (\frac{φ′}{r} v, v)_{L^{2}} (6.25)

Compute

I = ∥(P_{φ} − e)v∥_{L^{2}}^{2} = ∥Re(P_{φ} − e)v∥_{L^{2}}^{2} + ∥Im(P_{φ} − e)v∥_{L^{2}}^{2} + i ([ReP_{φ}, ImP_{φ}]v, v)_{L^{2}}
= (A) + (B) + (C)

(6.26)

(B) = ∥2φ′h∂_{r}v + hφ′′v∥_{L^{2}}^{2} ≥ 2∥φ′h∂_{r}v∥_{L^{2}}^{2} − h^{2}∥φ′′v∥_{L^{2}}^{2} (6.27)

\begin{aligned}(C) & = ((−4h^{2}∂_{r}φ′′h∂_{r} − 4h\frac{φ′}{r} \frac{h^{2}Δ_{σ}}{r^{2}} + 4h(φ′)^{2}φ′′v, v)_{L^{2}} \\ & + ((−h^{3}∂_{r}φ′′′ + h^{3}φ′′′∂_{r} − 2hφ′∂_{r}V)v, v)_{L^{2}} \\ & = (1) + (2) + (3) + (4) + (5) + (6) \end{aligned} (6.28)

(6.29)  (1) = 4h (φ′′h∂_{r}v, h∂_{r}v)_{L^{2}}
(6.30)  (2) = 4h (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}}
(6.31)  (3) = 4h ((φ′)^{2}φ′′v, v)_{L^{2}}
(6.32) |(4) + (5) + (6)| ≤ h^{2} ((|φ′′′|h∂_{r}v, h∂_{r}v)_{L^{2}} + (|φ′′′|v, v)_{L^{2}}) + 2h (φ′|∂_{r}V |v, v)_{L^{2}}

We need the following lemma:

Lemma 6.3. — There exists h_{0} > 0 such that for any 0 < h < h_{0}

(6.33) (\frac{φ′}{r} v, v)_{L^{2}} ≤ 2 (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}}

+ 2 (\frac{φ′}{r} h∂_{r}v, h∂_{r}v)_{L^{2}} + 3∥ (\frac{φ′}{r} )^{1/2} (P_{φ} − e)v∥_{L^{2}}^{2}

32 NICOLAS BURQ

Proof of Lemma 6.3: Compute

\begin{aligned}Re & ((P_{φ} − e)v, \frac{φ′}{r} v)_{L^{2}} \\ & = (\frac{φ′}{r} h∂_{r}v, h∂_{r}v)_{L^{2}} + (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} \\ & − (\frac{φ′}{r} (e + (φ′)^{2} − V)v, v)_{L^{2}} + hRe ((\frac{φ′}{r} )′ h∂_{r}v, v)_{L^{2}} \\ & − h (φ′ (\frac{φ′}{r} )′ v, v)_{L^{2}} \\ & = (\frac{φ′}{r} h∂_{r}v, h∂_{r}v)_{L^{2}} + (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} \\ & − ([\frac{φ′}{r} (e + (φ′)^{2} − V) + hφ′ (\frac{φ′}{r} )′ + \frac{h^{2}}{2} (\frac{φ′}{r} )′′] v, v)_{L^{2}} \end{aligned} (6.34)

From proposition 4.13, we obtain

(6.35) Re ((P_{φ} − e)v, \frac{φ′}{r} v)_{L^{2}} = (\frac{φ′}{r} h∂_{r}v, h∂_{r}v)_{L^{2}} + (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}}

− (\frac{φ′}{r} (e + (φ′)^{2} − V + O(h^{1/2})) v, v)_{L^{2}}

but

(6.36) |Re ((P_{φ} − e)v, \frac{φ′}{r} v)_{L^{2}} | ≤ ∥ (\frac{φ′}{r} )^{1/2} (P_{φ} − e)v∥_{L^{2}} × ∥ (\frac{φ′}{r} )^{1/2} v∥_{L^{2}}

≤ \frac{3}{2}∥ (\frac{φ′}{r} )^{1/2} (P_{φ} − e)v∥_{L^{2}}^{2} + \frac{1}{6}∥ (\frac{φ′}{r} )^{1/2} v∥_{L^{2}}^{2}

From (4.3) and (6.36) we obtain lemma 6.3.

We come back to the proof of proposition 6.2. From (6.27), (6.30) and lemma 6.3, we get

(6.37) (B) + (2) + 3h ∥(\frac{φ′}{r} )^{1/2} (P_{φ} − e)v∥_{L^{2}}^{2}

≥ ([2(φ′)^{2} − 2h\frac{φ′}{r} ] h∂_{r}v, h∂_{r}v)_{L^{2}} + h (\frac{φ′}{r} (4 − 2)\frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}}

+ h ((\frac{φ′}{r} − h(φ′′)^{2}) v, v)_{L^{2}}

and we obtain from (6.26), (6.29), (6.31), (6.32),

(6.38) I + 3h∥ (\frac{φ′}{r} )^{1/2} (P_{φ} − e)v∥_{L^{2}}^{2} ≥ 2h (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}}

+ ([2(φ′)^{2} − 2h\frac{φ′}{r} + 4hφ′′ − h^{2}|φ′′′|] h∂_{r}v, h∂_{r}v)_{L^{2}}

+ h ([\frac{φ′}{r} + 4(φ′)^{2}φ′′ − h(φ′′)^{2} − h|φ′′′| − 2φ′|∂_{r}V |] v, v)_{L^{2}}

From lemma 4.10, (4.3), proposition 4.11 and proposition 4.12 we obtain

(6.39) I + 3h∥ (\frac{φ′}{r} )^{1/2} (P_{φ} − e)v∥_{L^{2}}^{2}

≥ ((φ′)^{2}h∂_{r}v, h∂_{r}v)_{L^{2}} + 2h (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} + \frac{h}{2} (\frac{φ′}{r} v, v)_{L^{2}}

proving proposition 6.2.

6.3. The dissipation-commutator estimate. — In this section we show how propositions 6.1 and 6.2 can be mixed together to give an estimate on solution to the analitically dilated problem.

Proposition 6.4 (Proposition 5.2). — There exists C > 0 and h_{0} > 0 such that for any 0 < h < h_{0} and any u ∈ D_{θ_{0}}, with v = χ_{2}e^{φ/h}u

(6.40) C∥χ_{2}e^{φ/h}(P_{θ_{0}} − e)u∥_{L^{2}}^{2} − Ce^{2φ(B,h)/h}Im (e^{iκ_{θ_{0}}(r)}(P_{θ_{0}} − e)u, u)_{L^{2}}

≥ ((φ′)^{2}h∂_{r}v, h∂_{r}v)_{L^{2}} + h (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} + h (\frac{φ′}{r} v, v)_{L^{2}}

+ e^{2φ(B,h)/h} ∫ κ_{θ_{0}}(r) (|h∇u|^{2} + |u|^{2}) − Ch^{2} ∫_{A≤r≤A+1} e^{2φ/h}(|u|^{2} + |h∇u|^{2})

From proposition 6.1

(6.41) − Im (e^{iκ_{θ_{0}}(r)}(P_{θ_{0}} − e)u, u)_{L^{2}}

≥ \frac{1}{2} ∫ κ_{θ_{0}}(r)(|h∇u|^{2} + |u|^{2}) − ε(δ)h^{2} ∫_{B≤r≤B+δ} |u|^{2}

From proposition 6.2

2∥(P_{φ} − e)v∥^{2} ≥ ∥φ′h∂_{r}v∥_{L^{2}}^{2} + h (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} + \frac{h}{2} (\frac{φ′}{r} v, v)_{L^{2}} (6.42)

34 NICOLAS BURQ

But

\begin{aligned}(P_{φ} − e)v & = e^{φ/h}(P_{θ_{0}} − e)χ_{2}u + e^{φ/h}(P − P_{θ_{0}})χ_{2}u \\ & = e^{φ/h}χ_{2}(P_{θ_{0}} − e)u + e^{φ/h}[P_{θ_{0}}, χ_{2}]u + e^{φ/h}(P − P_{θ_{0}})χ_{2}u \\ & = (1) + (2) + (3) \end{aligned} (6.43)

Since for r ≥ B, we have φ(r, h) = φ(B, h) + O(h), we obtain (6.44) ∥(2)∥_{L^{2}}^{2} ≤ Ch^{2} ∫_{A≤r≤A+1/2} e^{2φ/h}(|h∇u|^{2} + |u|^{2})drdL

+ Ch^{2}e^{2φ(B,h)/h} ∫_{B+1/2≤r≤B+1}(|h∇u|^{2} + |u|^{2})drdL

Since the coefficients of the operator (P − P_{θ_{0}}) are equal to 0 for r ≤ B, it is easy to see from (6.10) that

∥(3)∥_{L^{2}}^{2} ≤ Ce^{2φ(B,h)/h} ∫ (|κ_{θ_{0}}(r)| + |κ_{θ_{0}}(r)′| + |κ_{θ_{0}}(r)′′|)^{2} ∑_{|α|≤2} |(h∂_{x})^{α}χ_{2}u|^{2} (6.45)

Lemma 6.5. — There exists C > 0 and h_{0} > 0 such that for any 0 < h < h_{0}, u ∈ D_{θ_{0}},

∥(3)∥_{L^{2}}^{2} ≤ Ce^{2φ(B,h)/h} ∫ κ_{θ_{0}}(r) ∑_{|α|≤1} |(h∂_{x})^{α}u|^{2} + κ_{θ_{0}}(r)|χ_{2}(P_{θ_{0}} − e)u|^{2} (6.46)

Proof of lemma 6.5 P_{θ_{0}}(χ_{2}κ_{θ_{0}}(r)u) = χ_{2}κ_{θ_{0}}(r)(P_{θ_{0}} − e)u + [P_{θ_{0}}, χ_{2}κ_{θ_{0}}(r)]u + eχ_{2}κ_{θ_{0}}(r)u (6.47) By standard semi-classical elliptic estimates for the operator P_{θ_{0}}, we obtain

(6.48) ∑_{|α|≤2} ∥(h∂_{x})^{α}χ_{2}κ_{θ_{0}}(r)u∥_{L^{2}}^{2} ≤ C(∥χ_{2}κ_{θ_{0}}(r)(P_{θ_{0}} − e)u∥_{L^{2}}^{2}

+ ∥eχ_{2}κ_{θ_{0}}(r)u∥_{L^{2}}^{2} + ∥[P_{θ_{0}}, χ_{2}κ_{θ_{0}}(r)]u∥_{L^{2}}^{2})

Hence

(6.49) ∑_{|α|=2} ∥κ_{θ_{0}}(r)(h∂_{x})^{α}χ_{2}u∥_{L^{2}}^{2} ≤ C(∥χ_{2}κ_{θ_{0}}(r)(P_{θ_{0}} − e)u∥_{L^{2}}^{2}

+ ∑_{|α|≤1} ∥|(κ_{θ_{0}}(r)| + |κ_{θ_{0}}(r)′| + |κ_{θ_{0}}(r)′′|)(h∂_{x})^{α}u∥_{L^{2}}^{2})

By the same way we get similar estimates for

∑_{|α|=2} ∥κ′_{θ_{0}}(r)(h∂_{x})^{α}χ_{2}u∥_{L^{2}}^{2}, and ∑_{|α|=2} ∥κ′′_{θ_{0}}(r)(h∂_{x})^{α}χ_{2}u∥_{L^{2}}^{2}, (6.50)

which gives

(6.51) ∑_{|α|=2} ∥(|κ_{θ_{0}}(r)| + |κ_{θ_{0}}(r)′| + |κ_{θ_{0}}(r)′′|)(h∂_{x})^{α}χ_{2}u∥_{L^{2}}^{2}

≤ C[∥χ_{2}(|κ_{θ_{0}}(r)| + |κ_{θ_{0}}(r)′| + |κ_{θ_{0}}(r)′′|)(P_{θ_{0}} − e)u∥_{L^{2}}^{2}

+ ∑_{|α|≤1} ∥(κ_{θ_{0}}(r)| + |κ_{θ_{0}}(r)′| + |κ_{θ_{0}}(r)′′| + |κ_{θ_{0}}(r)′′′| + |κ_{θ_{0}}(r)^{(4)}|)(h∂_{x})^{α}u∥_{L^{2}}^{2}

According to (2.18) we obtain lemma 6.5.

We come back to the proof of proposition 6.4. Take N ∈ ℕ to be fixed later. From (6.41), (6.43) and (6.42),

(6.52) I = 2∥e^{φ/h}χ_{2}(P_{θ_{0}} − e)u∥^{2} − Ne^{2φ(B,h)/h}Im (e^{iκ_{θ_{0}}(r)}(P_{θ_{0}} − e)u, u)_{L^{2}}

≥ ∥(P_{φ} − e)v∥^{2} − 4[∥(2)∥^{2} + ∥(3)∥^{2}]

+ Ne^{2φ(B,h)/h} [\frac{1}{2} ∫ κ_{θ_{0}}(r)(|h∇u|^{2} + |u|^{2}) − ε(δ)h^{2} ∫_{B≤r≤B+δ} |u|^{2}]

From (6.42), (6.44)and lemma 6.5 we obtain

\begin{aligned}I & + Ce^{2φ(B,h)/h} ∫ κ_{θ_{0}}(r)|(χ_{2}P_{θ_{0}} − e)u|^{2} \\ & ≥ \frac{1}{2}∥φ′h∂_{r}v∥_{L^{2}}^{2} + \frac{h}{2} (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} + \frac{h}{4} (\frac{φ′}{r} v, v)_{L^{2}} \\ & + Ne^{2φ(B,h)/h} [\frac{1}{2} ∫ κ_{θ_{0}}(r)(|h∇u|^{2} + |u|^{2})] \\ & − Ch^{2} ∫_{A≤r≤A+1/2} e^{2φ/h}(|u|^{2} + |h∇u|^{2}) \\ & − Nε(δ)h^{2}e^{2φ(B)/h} ∫_{B≤r≤B+δ} |u|^{2} \\ & − Ch^{2}e^{2φ(B,h)/h} ∫_{B+1/2≤r≤B}(|u|^{2} + |h∇u|^{2}) \\ & − Ce^{2φ(B,h)/h} ∫ κ_{θ_{0}}(r)(|u|^{2} + |h∇u|^{2}) \end{aligned} (6.53)

To get proposition 6.4, we have to eliminate in (6.53) the three last terms (terms (6),

(7) and (8). First fix N ≥ 2(C + 1) Then

(6.54) (4) + (8) = Ne^{2φ(B,h)/h} [\frac{1}{2} ∫ κ_{θ_{0}}(r)(|h∇u|^{2} + |u|^{2})]

− Ce^{2φ(B,h)/h} ∫ κ_{θ_{0}}(r)(|u|^{2} + |h∇u|^{2})

≥ e^{2φ(B,h)/h} ∫ κ_{θ_{0}}(r)(|u|^{2} + |h∇u|^{2})

Then choose 0 < δ < 1/2 such that for 0 < h < h_{0}

Nε(δ)e^{2φ(B)/h}h^{2} ≤ \frac{1}{8} inf_{r∈[B,B+1/2]} h\frac{φ′}{r} e^{2φ(r)/h} (6.55)

⇔ ε(δ) ≤ \frac{1}{8N} inf_{r∈[B,B+1/2]} e^{2 \frac{(φ(r,h)−φ(B,h))}{h}} \frac{g′(r)}{r} (6.56)

with φ′ = hg′ = \frac{8h}{r} near B (g is as in lemma 4.9). Such a choice is clearly possible since for r ≥ B we have φ(r, h) = φ(B, h) + 8h log(\frac{r}{B}) and lim_{δ→0} ε(δ) = 0.

Since v = e^{φ/h}u for B ≤ r ≤ B + 1/2, from (6.55) we obtain

(6.57) (3) + (6) = \frac{h}{4} (\frac{φ′}{r} v, v)_{L^{2}} − Nε(δ)h^{2}e^{2φ(B,h)/h} ∫_{B≤r≤B+δ} |u|^{2}

≥ \frac{h}{8} (\frac{φ′}{r} v, v)_{L^{2}}

Finally, since for B ≤ r ≤ B + 1/2, we have κ_{θ_{0}}(r) ≥ c > 0, we obtain, using (6.54)

(6.58) (4) + (8) + (7) ≥= e^{2φ(B,h)/h} ∫ κ_{θ_{0}}(r)(|u|^{2} + |h∇u|^{2}) + (7)

≥ (1 − \frac{C}{c} h^{2})e^{2φ(B,h)/h} ∫ κ_{θ_{0}}(r)(|u|^{2} + |h∇u|^{2})

With these choices of N, δ we obtain proposition 6.4.

7. The general case

In this section we come back to the general case. We work here in the coordinate system and after the conjugations of section 2.2. Since all the computations will be performed on functions supported in the set {|x| ≥ A}, the use of these reductions is licit. As in section 6, take R + 1 ≤ A ≤ R_{3} ≤< B and suppose that R is large enough so that for r ≥ R the operator P satisfies (2.12), (2.15) and (2.13). We note as in section 6 P_{θ_{0}} the operator obtained from P by the analytic dilation process presented in section 2.3. The point in this section is to prove the analogues of propositions 6.1 and 6.2. Unfortunatly, we can not obtain these propositions (in the general case) by a simple perturbation argument from the results obtained in the potential case, but more likelly from a perturbation of their proofs.

7.1. Dissipation estimates. —

Proposition 7.1. — For any δ > 0 there exists h_{0} such that for any 0 < h < h_{0} and u ∈ D_{θ_{0}}

(7.1) − Im (e^{iκ_{θ_{0}}} (P_{θ_{0}} − e)u, u)_{L^{2}}

≥ \frac{1}{2} ∫ (κ_{θ_{0}} + rκ′_{θ_{0}})|h∂_{r}u|^{2} + κ_{θ_{0}}(|\frac{h∇_{σ}}{r} u|^{2} + |u|^{2})drdL − h^{2}ε(δ) ∫_{B≤r≤B+δ} |u|^{2}

LOWER BOUNDS FOR SHAPE RESONANCES 37

with

lim_{δ→0} ε(δ) = 0 (7.2)

As in section 6 to prove this result, since the operator P is self adjoint,we can suppose that the function u is supported in the set {|x| ≥ A}. We have

(7.3) P_{θ_{0}} = − (\frac{e^{−iκ_{θ_{0}}}}{1 + irκ′_{θ_{0}}} h∂r)^{2} − h^{2}e^{−2iκ_{θ_{0}}} \frac{Δ_{σ}}{r^{2}}

+ h^{2}e^{−2iκ_{θ_{0}}} q_{2}(re^{iκ_{θ_{0}}} , σ, \frac{h∂_{σ_{i}}}{r}) + e^{−iκ_{θ_{0}}} q_{1}(re^{iκ_{θ_{0}}} , σ, \frac{h∂_{σ_{i}}}{r} , h) + V (re^{iκ_{θ_{0}}} , σ, h)

and (7.4) − Im (e^{iκ_{θ_{0}}} (P_{θ_{0}} − e)u, u)_{L^{2}}

= ∫ Im (−\frac{e^{−iκ_{θ_{0}}}}{(1 + irκ′_{θ_{0}})^{2}} ) |h∂_{r}u|^{2} + Im ∫ −e^{−iκ_{θ_{0}}} \frac{∥∇_{σ}u∥^{2}}{r^{2}}

+ Im ∫ e^{−iκ_{θ_{0}}} q_{2}(r, σ, \frac{h∂_{σ_{i}}}{r})u\overline{u}drdL + ∫ Im (e^{iκ_{θ_{0}}} (e − V (re^{iκ_{θ_{0}}} , σ, h))) |u|^{2}drdL

− hIm (∫ g(r)h∂_{r}u\overline{u}drdL) + Im ∫ q_{1}(re^{iκ_{θ_{0}}} , σ, \frac{h∂_{σ_{i}}}{r})u\overline{u}drdL

+ Im ∫ e^{−iκ_{θ_{0}}} (q_{2}(re^{iκ_{θ_{0}}} , σ, \frac{h∂_{σ_{i}}}{r}) − q_{2}(r, σ, \frac{h∂_{σ_{i}}}{r})) u\overline{u}drdL

= (1) + (2) + (3) + (4) + (5) + (6) + (7)

with (from (2.19))

(1) ≥ \frac{3}{4} ∫ (κ_{θ_{0}} + rκ′_{θ_{0}})|h∂_{r}u|^{2}, (2) ≥ \frac{3}{4} ∫ κ_{θ_{0}} \frac{∥∇_{σ}u∥^{2}}{r^{2}} (7.5)

Since q_{2} is self adjoint, we have

(3) = ∫ Im(e^{−iκ_{θ_{0}}} )q_{2}(r, σ, \frac{h∂_{σ_{i}}}{r})u\overline{u} (7.6)

and from (4.3) and (4.4)

|(3)| ≤ \frac{1}{16} ∫ κ_{θ_{0}}(|h\frac{∇_{σ}}{r} u|^{2} + |u|^{2}) (7.7)

As in section 6, we obtain

(4) ≥ \frac{3}{4} ∫ κ_{θ_{0}}|u|^{2} (7.8)

and |(5)| ≤ C(δ)h ∫ κ_{θ_{0}}(|h∂_{r}u|^{2} + |u|^{2}) + ε(δ)h^{2} ∫_{B≤r≤B+δ} |u|^{2} (7.9)

38 NICOLAS BURQ

Remark that since the operator q_{1}(r, σ, \frac{h∂_{σ_{i}}}{r}) is self adjoint, we have (6) = Im(6′) with

(6′) = ∫ (q_{1}(re^{iκ_{θ_{0}}} , σ, \frac{h∂_{σ_{i}}}{r}) − q_{1}(r, σ, \frac{h∂_{σ_{i}}}{r})) u\overline{u} (7.10)

From (4.5) we deduce

|(6′)| ≤ \frac{1}{16} (∫ κ_{θ_{0}}|h\frac{∇_{σ}}{r} u|^{2} + ∫ κ_{θ_{0}}|u|^{2}) (7.11)

We also have from (4.5)

(7.12) |(7)| = | ∫ e^{−iκ_{θ_{0}}} (q_{2}(re^{iκ_{θ_{0}}} , σ, \frac{h∂_{σ_{i}}}{r}) − q_{2}(r, σ, \frac{h∂_{σ_{i}}}{r})) u\overline{u}|

≤ \frac{1}{16} (∫ κ_{θ_{0}}|h\frac{∇_{σ}}{r} u|^{2} + ∫ κ_{θ_{0}}|u|^{2})

From (7.5), (7.7), (7.8), (7.9), (7.11)and (7.12) we obtain (7.13) − Im (e^{iκ_{θ_{0}}} (P_{θ_{0}} − e)u, u)_{L^{2}}

≥ ∫ (\frac{3}{4} − C(δ)h) (κ_{θ_{0}} + rκ′_{θ_{0}})|h∂_{r}u|^{2} + (\frac{3}{4} − \frac{3}{16}) κ_{θ_{0}}(|\frac{h∇_{σ}}{r} u|^{2}drdL

+ (\frac{3}{4} − \frac{3}{16} − C(δ)h) |u|^{2})drdL − h^{2}ε(δ) ∫_{B≤r≤B+δ} |u|^{2}drdL

For h_{0} small enough (depending on δ > 0), we obtain proposition 7.1 7.2. Commutator estimates. — Consider the function φ(r, h) constructed in section 4. Note

\begin{aligned}P_{φ} & = e^{φ/h}P e^{−φ/h} \\ & = −h^{2}∂_{r}^{2} − h^{2} \frac{Δ_{σ}}{r^{2}} + q_{2}(r, σ, h\frac{∂_{σ}}{r}) + q_{1}(r, σ, h\frac{∂_{σ}}{r} , h) + V − (φ′)^{2} + i(φ′h\frac{∂_{r}}{i} + h\frac{∂_{r}}{i} φ′) \\ & = ReP_{φ} + iImP_{φ} \end{aligned} (7.14)

Consider χ_{2} ∈ C_{0}^{∞}(ℝ^{d}) with support in {A ≤ r ≤ B + 1} and equal to 1 in {A + 1/2 ≤ r ≤ B + 1/2}. Proposition 7.2. — There exists h_{0} > 0 such that for any 0 < h < h_{0}, e ∈ [1, 2] and any u ∈ D_{θ}, if v = χ_{2}u

2∥(P_{φ} − e)v∥_{L^{2}}^{2} ≥ ∥φ′h∂_{r}v∥_{L^{2}}^{2} + h (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} + \frac{h}{2} (\frac{φ′}{r} v, v)_{L^{2}} (7.15)

To prove this result, we follow closely the proof of proposition 6.2. Compute

I = ∥(P_{φ} − e)v∥_{L^{2}}^{2} = ∥Re(P_{φ} − e)v∥_{L^{2}}^{2} + ∥Im(P_{φ} − e)v∥_{L^{2}}^{2} + i ([ReP_{φ}, ImP_{φ}]v, v)_{L^{2}}
= (A) + (B) + (C)

(7.16)

As in (6.27),

(B) ≥ 2∥φ′h∂_{r}v∥_{L^{2}}^{2} − h^{2}∥φ′′v∥_{L^{2}}^{2} (7.17)

(7.18) (C) = ((−4h^{2}∂_{r}φ′′h∂_{r} − h^{3}∂_{r}φ′′′ + φ′′′h^{3}∂_{r} − 4h\frac{φ′}{r^{3}} h^{2}Δ_{σ} + 4h(φ′)^{2}φ′′) v, v)_{L^{2}}

− ((2hφ′ \frac{d}{dr} (q_{2}(r, σ, h, h\frac{∂_{σ}}{r})) + 2hφ′ \frac{d}{dr} (q_{1}(r, σ, h, h\frac{∂_{σ}}{r}))) v, v)_{L^{2}} + 2h(φ′∂_{r}V v, v)_{L^{2}}

Taking into account that q_{2} is quadratic with respect to ∂_{σ}/r and that q_{1} is linear we obtain

\begin{aligned}(C) & = (4φ′′h∂_{r}v, h∂_{r}v)_{L^{2}} + 4h (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} + 4h ((φ′)^{2}φ′′v, v)_{L^{2}} \\ & + ((−h^{3}∂_{r}φ′′′ + h^{3}φ′′′∂_{r})v, v)_{L^{2}} − 2h (φ′(∂_{r}q_{2} − 2\frac{q_{2}}{r})v, v)_{L^{2}} \\ & − 2h (φ′(∂_{r}q_{1} − \frac{q_{1}}{r})v, v) − 2h (φ′∂_{r}V v, v)_{L^{2}} \\ & = (1) + (2) + (3) + (4) + (5) + (6) + (7) \end{aligned} (7.19)

The terms (1), (2) and (3) are the same as in (6.28) and we have as in (6.29), (6.31) and (6.32)

(7.20)  (1) = 4h (φ′′h∂_{r}v, h∂_{r}v)_{L^{2}}
(7.21)  (2) = 4h (\frac{φ′}{r} h\frac{∇_{σ}}{r} v, h\frac{∇_{σ}}{r} v)_{L^{2}}
(7.22)  (3) = h ((φ′)^{2}φ′′v, v)_{L^{2}}
(7.23) |(4) + (7)| ≤ h^{2} ((|φ′′′|h∂_{r}v, h∂_{r}v)_{L^{2}} + (|φ′′′|v, v)_{L^{2}}) + 2h (φ′|∂_{r}V |v, v)_{L^{2}}

From (4.4) we have

|(5) + (6)|) ≤ \frac{h}{8} (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} + \frac{h}{8} (\frac{φ′}{r} v, v)_{L^{2}} (7.24)

We need the following lemma:

Lemma 7.3. — There exists h_{0} > 0 such that for any 0 < h < h_{0}

(7.25) (\frac{φ′}{r} v, v)_{L^{2}} ≤ 2 (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}}

+ 2 (\frac{φ′}{r} h∂_{r}v, h∂_{r}v)_{L^{2}} + 3∥ (\frac{φ′}{r} )^{1/2} (P_{φ} − e)v∥_{L^{2}}^{2}

Proof of Lemma 7.3: The same calculus as in the proof of lemma 6.3 gives

\begin{aligned}Re & ((P_{φ} − e)v, \frac{φ′}{r} v)_{L^{2}} \\ & = (\frac{φ′}{r} h∂_{r}v, h∂_{r}v)_{L^{2}} + (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} \\ & + (\frac{φ′}{r} q_{2}(r, σ, \frac{h∇_{σ}}{r} v, v)_{L^{2}} + (\frac{φ′}{r} q_{1}(r, σ, \frac{h∂_{σ}}{r})v, v)_{L^{2}} \\ & − ([\frac{φ′}{r} (e + (φ′)^{2} − V) − hφ′ (\frac{φ′}{r} )′ − \frac{h^{2}}{2} (\frac{φ′}{r} )′′] v, v)_{L^{2}} \end{aligned} (7.26)

From (4.4) we have

(7.27) |(\frac{φ′}{r} q_{2}(r, σ, \frac{h∂_{σ}}{r})v, v)_{L^{2}} | + |(\frac{φ′}{r} q_{1}(r, σ, \frac{h∂_{σ}}{r})v, v)_{L^{2}} |

≤ \frac{1}{8} (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} + \frac{1}{8} (\frac{φ′}{r} v, v)_{L^{2}}

From (4.3), proposition 4.13 and(7.27) we obtain

(7.28) (\frac{7}{8} − Ch^{1/2}) (\frac{φ′}{r} v, v)_{L^{2}} ≤ (\frac{φ′}{r} (e + (φ′)^{2} − V − hφ′ (\frac{φ′}{r} )′ − \frac{h^{2}}{2} (\frac{(φ′)}{r})′′) v, v)_{L^{2}}

≤ |Re ((P_{φ} − e)v, \frac{φ′}{r} v)_{L^{2}} | + (\frac{φ′}{r} h∂_{r}v, h∂_{r}v)_{L^{2}}

+ \frac{9}{8} (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} + \frac{1}{8} (\frac{φ′}{r} v, v)_{L^{2}}

but

(7.29) |Re ((P_{φ} − e)v, \frac{φ′}{r} v)_{L^{2}} | ≤ ∥ (\frac{φ′}{r} )^{1/2} (P_{φ} − e)v∥_{L^{2}} × ∥ (\frac{φ′}{r} )^{1/2} v∥_{L^{2}}

≤ \frac{3}{2}∥ (\frac{φ′}{r} )^{1/2} (P_{φ} − e)v∥_{L^{2}}^{2} + \frac{1}{6}∥ (\frac{φ′}{r} )^{1/2} v∥_{L^{2}}^{2}

From (7.28) and (7.29) we obtain lemma 7.3.

We come back to the proof of proposition 7.2. From (7.17), (7.24) and lemma 7.3, we get

(7.30) (B) + (2) + 3h∥ (\frac{φ′}{r} )^{1/2} (P_{φ} − e)v∥_{L^{2}}^{2}

≥ ((2(φ′)^{2} − 2h\frac{φ′}{r})h∂_{r}v, h∂_{r}v)_{L^{2}} + h (\frac{φ′}{r} (4 − 2) \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}}

+ h ((\frac{φ′}{r} − h(φ′′)^{2}) v, v)_{L^{2}}

and we obtain from (7.20), (7.22), (7.23), (7.24) and (4.3)

(7.31) I + 3h∥ (\frac{φ′}{r} )^{1/2} (P_{φ} − e)v∥_{L^{2}}^{2} ≥ ((2(φ′)^{2} − 2h\frac{φ′}{r} + 4hφ′′ − h^{2}|φ′′′|)h∂_{r}v, h∂_{r}v)_{L^{2}}

+ h (\frac{φ′}{r} (4 − 2 − 1/8)\frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}}

+ h ((\frac{φ′}{r} (1 − 1/8) + (φ′)^{2}φ′′ − h(φ′′)^{2} − h|φ′′′| − 2φ′|∂_{r}V |) v, v)_{L^{2}}

From lemma 4.10, proposition 4.11 and proposition 4.12 we obtain

(7.32) I + 3h∥ (\frac{φ′}{r} )^{1/2} (P_{φ} − e)v∥_{L^{2}}^{2}

≥ ((φ′)^{2}h∂_{r}v, h∂_{r}v)_{L^{2}} + \frac{15h}{8} (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} + \frac{h}{2} (\frac{φ′}{r} v, v)_{L^{2}}

proving proposition 7.2.

7.3. The dissipation-commutator estimate. — As in section 6.3, propositions 7.1 and 7.2 can be mixed together to give an estimate on solution to the analitically dilated problem.

Proposition 7.4 (Proposition 5.2). — There exists C > 0 and h_{0} > 0 such that for any 0 < h < h_{0} and any u ∈ D_{θ_{0}}, with v = χ_{2}e^{φ/h}u

(7.33) C∥χ_{2}e^{φ/h}(P_{θ_{0}} − e)u∥_{L^{2}}^{2} − Ce^{2φ(B)/h}Im (e^{iκ_{θ_{0}}} (P_{θ_{0}} − e)u, u)_{L^{2}}

≥ ((φ′)^{2}h∂_{r}v, h∂_{r}v)_{L^{2}} + h (\frac{φ′}{r} \frac{h∇_{σ}}{r} v, \frac{h∇_{σ}}{r} v)_{L^{2}} + h (\frac{φ′}{r} v, v)_{L^{2}}

+ e^{2φ(B)/h} ∫ κ_{θ_{0}} (|h∇u|^{2} + |u|^{2}) drdL − Ch^{2} ∫_{A≤r≤A+1} e^{2φ/h}(|u|^{2} + |h∇u|^{2})drdL

8. Decay for the local energy of solution of the wave equation

In this section we show how theorem 2 can be deduced from theorem 3 for the estimate (1.5) and from theorem 4 for the estimate (1.6).

Note P_{0} = Δ_{g} the Laplace operator associated to the metric g. Note P the operator −h^{2}P_{0} and P_{θ_{0}}, the operator obtained from P by analytic dilation (for suitable θ_{0} and B >> 1).

From theorem 3 (with e = 1), we obtain

∥(P_{θ_{0}} − e)^{−1}∥_{L^{2}(Γ_{θ_{0}})} ≤ Ce^{C′/h}, e = 1 (8.1)

By a perturbation argument, estimate (8.1) is stil valid (with a different C) for |Ime| ≤ \frac{C^{−1}}{2} e^{−C′/h}. Remark that if χ ∈ C_{0}^{∞}(ℝ^{d}) is supported in the set \{|x| ≤ B\}, (the analytic dilation has been performed for r ≥ B), then χ(P_{θ_{0}} − e)^{−1}χ is the truncated outgoing resolvent of the operator P. This remark and (8.1) gives

∥χ(P − e)^{−1}χ∥_{L^{2}(Γ_{θ_{0}})} ≤ Ce^{C′/h}, |Ime| ≤ \frac{C^{−1}}{2} e^{−C′/h} (8.2)

Coming back by scaling to the resolvent for the operator P_{0}, we obtain the following estimate for the outgoing resolvent of the operator P_{0}, R(λ) = (P_{0} − λ^{2})^{−1} (with different C, C′).

Proposition 8.1. — There exists λ_{0} > 0, C, C′ such that the outgoing resolvent of the operator P_{0} is holomorphic in the set

\{λ ∈ ℂ ; Imλ < \frac{1}{C} e^{−C′|Reλ|}\} ∩ \{|λ| > λ_{0}\}. (8.3)

and satisfies there the estimate:

∥χR (λ) χ∥_{L(L^{2};L^{2})} ≤ Ce^{C′|Reλ|}. (8.4)

The same analysis as in the appendix B of [Bur98] shows that χR (λ) χ has no pole for λ real, 0 < |λ| ≤ λ_{0} (and hence, by compacity in a neighborhood of this set except close to 0) and that it is bounded close to 0 in any angular sector Λ, disjoint from the imaginary axis:

Λ = {λ ∈ ℂ^{∗}; arg(λ) ∈ [−θ_{1}, θ_{1}}; 0 < θ_{1} < π/2, |λ| < ε << 1} (8.5)

with these estimates we are in position to apply [Bur98, Theorem3] which gives (1.5).

For the proof of (1.6), the same analysis (and scaling) as above implies that if χ ∈ C_{0}^{∞}(ℝ^{d}), is supported in the set {R_{1} ≤ |x| ≤ R_{2}} and equal to 1 in the set {R_{1} + 1/2 ≤ |x| ≤ R_{2} − 1/2} then the outgoing resolvent R(λ) = (P_{1} − λ^{2})^{−1} satisfies the following estimate

∥χ(P_{0} − λ^{2})^{−1}χ∥_{L(L^{2};L^{2})} ≤ \frac{C}{|λ|} (8.6)

Take χ̃ ∈ C_{0(ℝ^{d})}^{∞} supported in the set where χ is equal to 1 and equal to 1 in the set {R_{1} + 2/3 ≤ |x| ≤ R_{2} − 2/3}.

Remark that (8.6) implies ,

∥χ̃(P_{0} − λ^{2})^{−1}χ̃∥_{L(L^{2};H^{1})} ≤ C (8.7)

We recall the classical equality (see [Bur98, (0.2)])

(P_{0} − λ^{2})^{−1} = − ∫_{0}^{+∞} e^{itλ} \frac{sin(\sqrt{−P_{1}})}{\sqrt{−P_{1}}} (8.8)

(As compared to [Bur98, (0.2)], the minus sign before the integral appears since here we consider P_{0} − λ^{2} instead of −P_{0} + λ^{2} and the minus sign in the exponent because according to our conventions the resonances here are the conjugate of the resonances in [Bur98]).

From (8.8) we obtain

\frac{sin(t\sqrt{−P_{1}})}{\sqrt{−P_{1}}}1_{t>0} = \frac{1}{2π} ∫_{−∞}^{+∞} e^{−itλ}(P_{1} − λ^{2})^{−1} = F^{−1}((P_{1} − λ^{2})^{−1}) (8.9)

Suppose now that f ∈ L^{2} is supported in the set {|x| ∈ [R_{1}+2, R_{2}−2]}. By finite speed of propagation (and if R_{1} is large enough so that the speed is close to 1) \frac{sin(\sqrt{−P_{1}})}{\sqrt{−P_{1}}}f is for t ∈ [0, 1] supported in the set where χ = 1. Consider Ψ ∈ C^{∞}(ℝ) an increasing function equal to 0 for t ≤ 0 and to 1 for t ≥ 1. Note

u(t) = (Ψ(t)\frac{sin(t\sqrt{−P_{1}})}{\sqrt{−P_{1}}}f) (8.10)

From the equation

(∂_{t}^{2} − P_{1}) (Ψ(t)u(t, x)) = Ψ′′(t)u(t, x) + 2Ψ′(t)∂_{t}u(t, x) = g(t, x) (8.11)

and Duhamel formula we obtain

Ψ(t)u(t, x) = (\frac{sin(s\sqrt{−P_{1}})}{\sqrt{−P_{1}}} ⋆_{s} g(s)) (t) (8.12)

Remark that in (8.12) g(s) is bounded in L^{∞}(ℝ_{s}; L^{2}(Ω)) by ∥f∥_{L^{2}} and supported in [0, 1], hence is bounded in L^{2}(ℝ_{s}; L^{2}(Ω))by ∥f∥_{L^{2}}. Remark also that

χ̃Ψ(t)u(t, x) = χ (\frac{sin(s\sqrt{−P_{1}})}{\sqrt{−P_{1}}} χ̃ ⋆_{s} g(s)) (t) (8.13)

indeed, g(s) is, by the finite speed of propagation argument we mentionned above, supported (in the x variable) in the region where χ̃ is equal to 1. By taking Fourier transforms in (8.13) we obtain

F_{r↦λ} (χΨ(t)u(t, x)) (λ) = −χ(P_{1} − (−λ)^{2})^{−1}χF_{s↦λ}(g(s))(λ) (8.14)

From (8.7) we obtain that the right hand side in (8.14) is bounded in L^{2}(ℝ_{λ}; H^{1}(Ω)) by ∥f∥_{L^{2}}. By Parseval identity, we obtain that

∥χΨ(t)u(t, x)∥_{L^{2}(ℝ_{t};H^{1})} ≤ C∥f∥_{L^{2}} (8.15)

which is exactly (1.6) for initial data (u_{0}, u_{1}) = (0, f). The proof for initial data of the form (u_{0}, u_{1}) = (g, 0) is similar.

Appendix A Existence of a geodesic coordinate system at infinity

In this section we give an outline of the proof of proposition 2.2

Note Z = {z = (z_{1}, …, z_{d}) ∈ ℂ^{d}; ∑ z_{i}^{2} = 1} the complexified d − 1 sphere and 𝕊_{ε}^{d−1} the intersection of Z with an ε neighborhood of ℝ^{d} (in ℂ^{d}). The purpose of this part is the following result:

Proposition A.1. — ∃ε, θ_{1} > 0, R_{0} >> 1 and

χ : x = (rσ), (r, σ) ∈ R_{0}+]0, +∞[×e^{i]−θ_{1},θ_{1}[} × 𝕊_{ε}^{d−1} ↦ χ(x) ∈ Ω ⊂ ℂ^{d} (A.1)

a holomorphic diffeomorphism such that 1.

χ(]R_{0}, +∞[×𝕊_{ℝ^{d}}^{d−1}) = (]R_{0}, +∞[×𝕊_{ℝ^{d}}^{d−1}) (A.2)

2. Denoting χ_{r} and χ_{σ} the radial coordinates, we have

|\frac{χ_{r}(r, σ)}{r} − 1| ≤ \frac{C}{|r|^{β}} (A.3)

and

|χ_{σ}(r, σ) − σ| ≤ \frac{C}{|r|^{β}} (A.4)

3. The principal symbol of the operator χ_{∗}(Q) is equal to

ϱ^{2} + q_{2}(r, σ, \frac{η}{r}) (A.5)

Remark A.2. — Remark that (A.3) and (A.4) imply that

\frac{∥χ(r, σ) − rσ∥}{|r|} ≤ \frac{C}{|r|^{β}} (A.6)

which means that the change of coordinate is asymptotically the identity. Proposition A.1 is nothing but a global result of a well known result: the existence of geodesic coordinate systems which allows, modulo the choice of a good coordinate system, to suppress the ∂_{r}∂_{σ} terms. Here two ingredients come into account: the global existence due to the global assumption on the asymptotic flatness and the analyticity property due to the analyticity at infinity assumption.

Since most of the work is standard estimates on solutions of differential equations, we will not give all the details of the (fastidious) proof of this result. We’ll emphasize on the global existence of the geodesic coordinate system.

Note

q(x, ξ) = ∑_{|α|=2} a_{α}(x)ξ^{α} (A.7)

the classical principal symbol of the operator Q. We work in polar coordinates and still note q(r, σ, ϱ, η) the function q in the polar (and their dual) coordinates. We recall that this function is a second order polynomial in the (ρ, η/r) variables with coefficients holomorphic with respect to (r, σ) in set of the form (with d > 0, R_{0} > 0)

Λ_{θ_{0},R_{0},d_{0}} = {(r, σ) ∈ ℂ^{d}; dist (σ, 𝕊^{d−1}) < d_{0}, r ∈ e^{i]−θ_{0},θ_{0}[}]R_{0}, +∞[} (A.8)

Consider the following system of differential equations:

\frac{d}{ds}r = \frac{1}{2}∂_{ρ}q, \frac{d}{ds}σ = \frac{1}{2}∂_{η}q, \frac{d}{ds}ρ = −\frac{1}{2}∂_{r}q, \frac{d}{ds}η = −\frac{1}{2}∂_{σ}q, (A.9)

with initial data

(A.10) r |_{s=0}= r_{0}, σ |_{s=0}= σ_{0} close to 𝕊_{ℝ^{d}}^{d−1} , ρ_{s=0} = ϱ_{0}, η_{s=0} = η_{0}

For any ε > 0 and R_{0} > 0, note

(A.11) D(ε, R_{0}) = {(r, σ, ϱ, η) ∈ ℂ^{2d}; |r − R_{0}| < ε, dist(σ, 𝕊_{ℝ^{d}}^{d−1}) < ε, |ϱ − 1| < ε, p(r, σ, ϱ, η) = 1}

Proposition A.3. — ∃θ_{1} > 0, ε > 0, R_{1}, ∀R_{0} > R_{1}, ∀(r, σ, ϱ, η) ∈ D(ε, R_{0}) the system (A.9), (A.10) has a unique solution defined for s ∈ e^{i[−θ_{1},θ_{1}]}ℝ^{+} which is holomorphic with respect to (s, r_{0}, σ_{0}, ϱ_{0}, η_{0})

Fix ε, R_{0} > 0. Then by standart results about differential equations, the system (A.9), (A.10) has a unique solution for s in a neighborhood of 0 in ℂ, for initial data in D(ε, R_{0}). We recall that, according to the asymptotic flatness assumption, we have ∀x ∈ Λ_{θ_{0},R_{0},d_{0}},

|q(r, σ, ρ, η) − (ϱ^{2} + \frac{|η|^{2}}{r^{2}} )| ≤ \frac{M}{|r|^{β}} (ϱ^{2} + \frac{|η|^{2}}{r^{2}} ) (A.12)

Furthermore, by Cauchy formula, (and possibly reducing θ_{0}), we have

|∂_{1}^{α_{1}}∂_{2}^{α_{2}}∂_{3}^{α_{3}}∂_{4}^{α_{4}}(p(r, σ, ρ, \frac{η}{r}) − (ϱ^{2} + \frac{|η|^{2}}{r^{2}} ))| ≤ \frac{M_{α}}{|r|^{β+α_{1}}} (ϱ^{2} + \frac{|η|^{2}}{r^{2}} ) (A.13)

Note I ⊂ ℝ^{+} the following set: s_{0} ∈ I if and only if for any (r, σ, ϱ, η) ∈ D(ε, R_{0}) the system (A.9), (A.10) has a unique solution defined for s ∈ [0, s_{0}]e^{i[−θ_{0}/2,θ_{0}/2]} and such that ∀s ∈ [0, s_{0}]e^{i[−θ_{0}/2,θ_{0}/2]}

\begin{matrix}|ṙ − 1|(s) ≤ \frac{1}{8} + \frac{M}{|r|^{β}(s)},  |ϱ̇|(s) ≤ \frac{M}{|r|^{β+1(s)}} \\ |σ̇(s)| ≤ \frac{M}{|r|^{β+1}(s)},  |η̇(s)| ≤ \frac{M}{|r|^{β}(s)} \end{matrix} (A.14)

Lemma A.4. — ∃ε_{0} > 0, R_{1} > 0 such that if 0 < ε < ε_{0} and R_{0} > R_{1} then I = ℝ^{+}.

To prove lemma A.4, we prove that I is non empty, closed and open in ℝ^{+}. I is clearly closed in ℝ^{+}, non empty since it contains a neighborhood of 0 in ℝ^{+}. To show that it is open, we first prove

Lemma A.5. — ∃ε_{0} > 0, R_{1} > 0 such that if 0 < ε < ε_{0} and R_{0} > R_{1}, then if s_{0} ∈ I, for any s ∈ [0, s_{0}]e^{i[−θ_{0}/2,θ_{0}/2]}, the point (r, σ, ϱ, η)(s) satisfies

(r, σ) ∈ Λ_{2θ_{0}/3,R_{1},d_{0}/2},  |ϱ − 1| ≤ \frac{1}{16},  |\frac{η}{r} | ≤ \frac{4}{3},  |r| ≥ R_{0} + \frac{|s|}{2} (A.15)

Proof: Fix s and consider for t ∈ [0, |s|],

t ↦ (r_{t}, σ_{t}, ϱ_{t}, η_{t}) = (r, σ, ϱ, η)(\frac{s}{|s|}t) (A.16)

We have from (A.14)

|\frac{d}{dt}r_{t} − \frac{s}{|s|}| ≤ \frac{1}{8} + \frac{M}{|r_{t}|^{β}} (A.17)

If R_{0} is taken large, then the right hand side in (A.17) is for t = 0 smaller than 1/4. Hence, |r_{t}| is increasing for small t and hence for all t and |\frac{d}{dt}r_{t} − \frac{s}{|s|}| ≤ \frac{1}{4}; which implies

\frac{3}{4} ≤ \frac{|r|}{|s| + R_{0}} ≤ \frac{5}{4} (A.18)

Using this inequality and (A.14), we obtain

|σ − σ_{0}| ≤ \frac{C}{R_{0}^{β}} , |η − η_{0}| ≤ C(R_{0} + |s|)^{1−β}, |ϱ − ϱ_{0}| ≤ \frac{C}{R_{0}^{β}} (A.19)

Hence

|\frac{η}{r} | ≤ \frac{C}{(R_{0} + |s|)^{β}} (A.20)

Comming back to (A.9), we obtain since ϱ^{2} + |\frac{η}{r} |^{2} is bounded from (A.12)

|ṙ − ϱ| ≤ \frac{C}{|r|^{β}} ≤ \frac{C}{(R_{0} + |s|)^{β}} (A.21)

and from (A.19) and (A.21)

|ṙ − 1| ≤ \frac{C}{R_{0}^{β}} + ε (A.22)

Now, if ε is small enough and R_{0} large enough, we obtain from (A.22) that r stays in the set e^{i[−2θ_{0}/3,2θ_{0}/3]}ℝ^{+} (because s ∈ e^{i[−θ_{0}/2,θ_{0}/2]}ℝ^{+}) and from (A.19) that σ stays close to 𝕊^{d−1}. The rest of the estimate (A.15) is now a simple consequence of (A.19), (A.20), (A.18) and of the inequality |ϱ_{0} − 1| ≤ ε.

We come back to the proof of lemma A.4. Suppose that s_{0} ∈ I. Consider z_{0} = s_{0}e^{iθ}, θ ∈ [−θ_{0}/2, θ_{0}/2]. According to lemma A.5, we have

(r, σ)(z_{0}) ∈ Λ_{\frac{2θ_{0}}{3} ,R_{0}, \frac{d_{0}}{2}} ,  |r|(z_{0}) ≥ R_{0} + \frac{s_{0}}{2} > R_{0},  |ϱ(z_{0})| ≤ \frac{17}{16} (A.23)

We can solve the equations (A.9),with initial data (r, σ, ϱ, η)(z_{0}) at z_{0} for z close to z_{0}, with a solution satisfying

\begin{matrix}|ṙ − ϱ| ≤ \frac{M}{|r|^{β}} ,  |σ̇| ≤ \frac{M}{|r|^{1+β}} \\ |ϱ̇| ≤ \frac{M}{|r|^{1+β}} ,  |η̇| ≤ \frac{M}{|r|^{β}} \end{matrix} (A.24)

as long as the solution (r, σ)(z) stays in Λ_{θ_{0},R_{0},d_{0}} (and at least for |z − z_{0}| small). Since |ϱ − 1|(z_{0}) ≤ \frac{1}{16}, for |z − z_{0}| small enough, we have |ϱ − 1|(z_{0}) ≤ \frac{1}{8} and we deduce that the solution of (A.9) with initial data (r, σ, ϱ, η)(z_{0}) at z_{0} exists and satisfies (A.14) for |z −z_{0}| small enough. This shows that the solution can be prolonged near z_{0}, satisfying (A.14). Using that {s_{0}} × e^{i[−θ_{0}/2,θ_{0}/2]} is compact, we obtain that the set I is open in ℝ^{+}. This concludes the proof of lemma A.4.

Using the equation satisfied by the differential of the application (r_{0}, σ_{0}, ρ_{0}, η_{0}) ↦ (r, σ, ρ, η) (s), we get (using (A.12), (A.13), (A.18) and (A.20))

\begin{aligned}|\frac{d}{ds}dr| & ≤ \frac{C}{(R_{0} + |s|)^{1+β}} |dr| + \frac{C}{(R_{0} + |s|)^{β}} |dσ| \\ & + C |dρ| + \frac{C}{(R_{0} + |s|)^{1+β}} |dη| \\ |\frac{d}{ds}dσ| & ≤ \frac{C}{(R_{0} + |s|)^{2+β}} |dr| + \frac{C}{(R_{0} + |s|)^{1+β}} |dσ| \\ & + \frac{C}{(R_{0} + |s|)^{1+β}} |dρ| + \frac{C}{(R_{0} + |s|)^{2+β}} |dη| \\ |\frac{d}{ds}dρ| & ≤ \frac{C}{(R_{0} + |s|)^{2+β}} |dr| + \frac{C}{(R_{0} + |s|)^{1+β}} |dσ| \\ & + \frac{C}{(R_{0} + |s|)^{1+β}} |dρ| + \frac{C}{(R_{0} + |s|)^{2+β}} |dη| \\ |\frac{d}{ds}dη| & ≤ \frac{C}{(R_{0} + |s|)^{1+β}} |dr| + \frac{C}{(R_{0} + |s|)^{β}} |dσ| \\ & + \frac{C}{(R_{0} + |s|)^{β}} |dρ| + \frac{C}{(R_{0} + |s|)^{1+β}} |dη| \end{aligned} (A.25)

differentiating the equation

q(r, σ, ϱ, η) = q(R_{0}, σ_{0}, ϱ_{0}, η_{0}) (A.26)

and using that ϱ is close to 1, we obtain

|dϱ| ≤ \frac{C}{(R_{0} + |s|)^{1+β}} |dr| + \frac{C}{(R_{0} + |s|)^{β}} |dσ| + \frac{C}{(R_{0} + |s|)^{1+β}} |dη| (A.27)

which allows to eliminate the |dϱ| terms in the right hand side of (A.25). Note

(A.28) f(s) = \frac{|dr − Id|^{2}}{(R_{0} + |s|)^{2−β}} + \frac{|dη − Id|^{2}}{(R_{0} + |s|)^{2−β}} + |dσ − Id|^{2} + |dρ − Id|^{2}

From (A.25) and(A.27) we obtain

|\frac{d}{ds}f| ≤ \frac{C}{(R_{0} + |s|)^{1−β/2}} + \frac{C}{(R_{0} + |s|)^{1−β/2}} (A.29)

and we obtain by Gronwall inequality

f(s) ≤ \frac{C}{(R_{0} + |s|)^{1−β/2}} (A.30)

For any σ_{0} ∈ 𝕊_{ε}^{d−1} note ϱ_{0} the unique solution (defined by the implicit function theorem) close to 1 of the equation q(R_{0}, σ_{0}, ϱ_{0}, η_{0}) = 1 which exists (and is holomorphic) if ε > 0 is small and R_{0} large enough. Using (A.12), (A.13), it is easy to see that

|ϱ − 1| ≤ \frac{C}{(R_{0})^{β}} , |∂_{σ_{0}}ϱ_{0}| ≤ \frac{C}{(R_{0})^{β}} (A.31)

Now consider the following map:

(A.32) χ_{0} : (t, σ_{0}) ∈ R_{0}+]0, +∞[×e^{i[−θ_{0}/2β,θ_{0}/2β]} × (𝕊^{d−1} + B_{ℂ^{d}}(0, ε))

↦ (r^{β/2}((t − R_{0}^{β/2})^{2/β}), σ((t − R_{0}^{β/2})^{2/β}))

where (r, σ, ϱ, η)(s) is the solution of (A.9) with initial data (R_{0}, σ_{0}, ϱ_{0}(σ_{0}), η_{0} = 0). The differential of the map χ_{0} is

dχ_{0} = (\begin{matrix}r^{β/2−1} × (t − R_{0}^{β/2})^{2/β−1}∂_{s}r & r^{β/2−1} (∂_{σ_{0}}r + ∂_{σ_{0}}ϱ_{0}∂_{ϱ_{0}}r) \\ _{\frac{2}{β(t−R_{0}^{β/2})^{2/β−1}}} ∂_{s}σ & ∂_{σ_{0}}σ + ∂_{σ_{0}}ϱ_{0}∂_{ϱ_{0}}σ \end{matrix}) (A.33)

Hence, from (A.30) and(A.31), we have

|dχ_{0} − Id| ≤ \frac{C}{(R_{0})^{β/2}} (A.34)

On the other hand, from (A.18), (A.20) and the fact that q = 1, we deduce

|ρ − 1| ≤ \frac{C}{(R_{0} + |s|)^{β}} (A.35)

From (A.25) we deduce

|ṙ − 1| ≤ \frac{C}{(R_{0} + |s|)^{β}} (A.36)

Note g(s) = r^{β/2}(s^{2/β}) − (s − R_{0}^{β/2}. From (A.36) and Gronwall inequality we deduce

|g|(s) ≤ \frac{C}{(R_{0} + |s|^{2/β})^{β/2}} (A.37)

This is equivalent to the fact that the r-component of χ_{0} is close to the identity:

|χ_{0,r}(t, σ) − t| ≤ \frac{C}{(R_{0} + |s|)^{β/2}} (A.38)

From (A.14) we have that the σ component of χ_{0} is also close to the identity.

Now, the map χ_{0} and its derivative are close to the identity (if R_{0} is taken large), so χ_{0} is a global diffeomorphism onto its image. Coming back to the map

χ̃ : R_{0}+]0, +∞[e^{i[−θ_{0}/2β,θ_{0}/2β]} × (𝕊_{ℝ^{d}}^{d−1} + B_{ℂ^{d}}(0, ε)) ↦ (r(s − R_{0}), σ(s − R_{0})) ⊂ ℂ^{d+1} (A.39)

we obtain that it is an holomorphic diffeomorphism onto its image.

Remark also that from (A.14), lim_{|s|→+∞} σ(s) exists. Indeed, we have

|σ(s, σ_{0}) − σ(s′, σ_{0})| ≤ | ∫_{s}^{\frac{|s′|}{|s|} s} σ̇| + | ∫_{\frac{|s′|}{|s|} s}^{s′} σ̇| ≤ \frac{M}{|s|^{β}} + \frac{M}{|s′|^{β}} (A.40)

It is again possible to show that the map Ψ : σ_{0} ↦ lim_{s→+∞} is an holomorphic diffeomorphism from 𝕊^{d−1} + B_{ℂ^{d}}(0, ε) onto a neighborhood of 𝕊^{d−1}, V in ℂ^{d}. Finally, the change of variables we are interested in, is the following (for ε small enough):

(A.41) χ : (r, σ) ∈ R_{0}+]0, +∞[×e^{i[−θ_{0}/2,θ_{0}/2]} × 𝕊_{ε}^{d−1}

↦ χ_{0}(r, Ψ^{−1}(σ))_{r} × χ^{−1}(r, σ)_{σ}

It is now easy to check that χ satisfies the desired asumptions.

A.1. Reductions for the operator P. — The operator P (seen in the new coordinate system) is self adjoint for the measure μ obtained from dx by the change of variables. Hence μ has the following form (dL is the Lebesgue measure on the d − 1dimensionnal sphere)

μ = f(r, σ)drdL (A.42)

with f(r, σ) a holomorphic function in a conic neighborhood of B(0, R_{0})^{c} satisfying there

|f(r, σ) − r^{d−1}| ≤ C|r|^{d−1−β/2} (A.43)

Note P_{1} = f^{1/2}P f^{−1/2}. It is easy to see that the operator P_{1} is self adjoint for the measure drdL and satisfies all the other assumptions [1,…, 4] of page 4 we have made about P. Notice that (since P_{1}^{∗} = P_{1}), we can suppose that the first order term in h∂_{r} is equal to

q_{1}(r, σ, h∂_{r}) = a_{1}(r, σ, h)h∂_{r} + h∂_{r}a_{1}(r, σ, h) (A.44)

with a_{1}(r, σ) ∈ iℝ; and that the 0 order term (the potential) is real.

Note

f̂(r, σ) = ∫_{z=R_{0}}^{r} a_{1}(r, σ)dz (A.45)

Then f̂ is purely imaginary and the operator P_{2} = e^{f̂/h}P_{1}e^{−f̂/h} is self adjoint for the measure drdL and has the following form

P_{2} = −h^{2}∂_{r}^{2} + q_{2}(r, σ, \frac{h∂_{σ}}{r}) + q̃_{1}(r, σ, \frac{h∇_{σ}}{r} , h) + V (r, σ, h) (A.46)

with different q̃_{1} and V, but still satisfying the assumptions [1,…, 4] of page 4 we have made about P.

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25th October 2000 Nicolas Burq, Université Paris Sud, Mathématiques, Bât 425, 91405, Orsay Cedex

E-mail : Nicolas.burq@math.u-psud.fr