Asymptotics of Resonances for Homoclinic Orbits

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Resonances in the semiclassical regime The Homoclinic case Sketch of proof (1 curve)

Asymptotics of Resonances for Homoclinic Orbits

Thierry Ramond LMO, Universit´e Paris Sud

after joint works with Jean-Fran¸cois Bony (Bordeaux 1), Setsuro Fujii´e (Ritsumeikan University) and Maher Zerzeri (Paris 13)

IHP, 04/11/2013

Thierry Ramond Asymptotics of Resonances for Homoclinic Orbits

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Outline

Resonances in the semiclassical regime Resonances

The semiclassical regime Some known results

The Homoclinic case The geometrical setting Main result

Some comments

Sketch of proof (1 curve)

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Resonances The semiclassical regime Some known results

Breit-Wigner peaks

IIn physics, the introduction of the notion of quantum resonance was motivated by the behavior of various quantities related to scattering experiments, as scattering cross sections, or time delays, . . .

a b

I At certain energies, these quantities present peaks, which were modelized by a Lorentzian shaped function

wa,b:λ7→ 1 π

b/2

(λ−a)²+ b/2².

I Note that forρ=a−ib/2∈C, one has

wa,b(λ) = 1 π

Imρ

|λ−ρ|²,

and the complex number ρwas called a resonance.

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From eigenvalues to resonances

ISuppose that a (1d) classical particle with energyE moves in one of the potential below. Its trajectory would be the same in both cases.

IFor a (1d) quantum particle, which is associated to anL²eigenfunction of the Schr¨odinger operator−h²∆ +V, the situation is drastically different: there could be noL² eigenfunction in the case at the right.

IHowever the existence of a ”classical trap” will give rise to the existence of resonant states and corresponding resonant energies, or resonances.

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The Schr¨ odinger operator

IWe consider the Schr¨odinger operator onRⁿ,n≥1,

P=−h²∆ +V(x) =Op^w_h(p) = 1 (2πh)ⁿ

Z Z

e^i(x−y^)·ξ/hpx+y 2 , ξ

dy dξ

wherep(x, ξ) =ξ²+V(x),V(x)∈C^∞(Rⁿ;R).

IFor the talk, we assume thatV ∈C₀^∞(Rⁿ;R).

IThenP=−h²∆ +V is self-adjoint onL²(Rⁿ) with domain H²(Rⁿ).

Its spectrum consists in negative eigenvalues (possibly), and in its essential spectrum [0,+∞[.

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The resolvent

ISinceP is self-adjoint, forImz >0 in particular, the bounded operator R(z) := (P−z)⁻¹:L²(Rⁿ)→L²(Rⁿ) is well defined.

Theorem and Definition:

The resolvent

{Rez >0,Imz >0} 3z

7→R(z) :L²_comp(Rⁿ)→L²_loc(Rⁿ)

can be meromorphicaly continued to the lower half-plane{Rez >0,Imz <0}.

Its poles are called resonances ofP, and we denoteRes(P) their set.

0 Im z

Re z σess(P)

σdisc(P)

(P-z)-1

R(z)

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Analytic distortion

ILetF:Rⁿ→Rⁿbe such thatF(x) = 0 for|x|<C andF(x) =x for

|x| 1.

Forµ∈Rsmall enough,Uµ:L²(Rⁿ)→L²(Rⁿ) is the unitary operator Uµϕ(x) =

det d xe^µF(x)

1/2ϕ xe^µF(x)

IForθ∈Rsmall enough, we denotePθ=UiθPU_iθ⁻¹the distorted operator.

The set of resonances ofP in Γθ={−2θ <ImE<0}is Resθ(P) =σ(Pθ)∩Γθ

Cⁿ

xe^iθF(x) Σ

Res(P) Γ_θ C

−2θ

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The truncated resolvent

IFor anyχ∈ C₀^∞(Rⁿ), the truncated resolventχR(z)χis well-defined as an operator onL²(Rⁿ), and the study ofR(z) can be reduced to that of its (suitably) truncated version.

IForχ∈C₀^∞(Rⁿ), the cut-off resolvent

χ(P−z)⁻¹χ:L²(Rⁿ)→L²(Rⁿ)

has a meromorphic extension to Γθ, and its poles are the resonances.

Moreover,

χ(P−z)⁻¹χ=χ(Pθ−z)⁻¹χ, whenF= 0 on suppχ.

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The trapped set

Definition:

The trapped set at energyE >0 is

K(E0) =

(x, ξ)∈p⁻¹(E0); t 7→exp(tHp)(x, ξ) is bounded

We recall thatp(x, ξ) =ξ²+V(x) is the semiclassical symbol of P=Op^w_h(p), and thatHpis the associated Hamiltonian vector field on T^∗Rⁿ=R²ⁿ, given byH_p=∂_ξp∂_x−∂_xp∂_ξ=

2ξ

−∇V(x)

.

IExamples:

- An empty trapped set.

- A hyperbolic trajectory.

- Le puits dans l’isle (shape resonances).

- A barrier top.

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The non-trapping case

Theorem (Martinez ’02)

Assume that K(E0) =∅. Then, P has no resonance in

Box(ε,Chlnh) = [E₀−ε,E₀+ε] +i[−Ch|lnh|,0]

for someε >0and all C >0. Moreover, for allχ∈C₀^∞(Rⁿ), there exists N=N(C)such that

kχ(P−z)⁻¹χk.h^−N in this set.

Moreover, ifV is analytic in a whole neighborhood ofRⁿof the form

Σ ={x∈Cⁿ; |Imx|< δhRexi}

then Briet–Combes–Duclos ’87 and Helffer–Sj¨ostrand ’86 (implicitly) have proved thatP has no resonance in a fixed neighborhood of E0.

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A hyperbolic closed trajectory

Theorem (C.G´ erard, Sj¨ ostrand ’87)

IfV is analytic in Σ, andK(E0) consists of a simple closed trajectory whose Poincar´e map is hyperbolic, then, inBox(ε,Ch), there is a bijectionbh

betweenRes(P) (counted with their multiplicity) and the set of roots of the equation

A(E) = 2kπh+ihlogρ(E)−ih

n−1

X

j=1

αjlogθj(E), k∈Z, αj ∈N

such thatbh(E) =E+o(h). HereA(E) is the action,θ1, . . . , θn−1 are the eigenvalues of the Poincar´e map with modulus>1 andρ(E) is some analytic function ofE satisfying

|ρ(E)|=|θ₁(E)· · ·θn−1(E)|⁻¹².

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A non-degenerate critical point

Theorem (Sj¨ostrand ’87)

SupposeV is analytic in Σ, andK(E0) consists of a non-degenerate critical point (0,0).

Denote

V(x) =E0−

d

X

j=1

λ²_j 4xj²+

n

X

j=d+1

µ²_j

4xj²+O(|x|³) as x→0 withE0>0,λj, µj>0, and

Eα,β=E0+h

n

X

j=d+1

µj(βj+1 2)−ih

d

X

j=1

λj(αj+1 2)

for each (α, β) = (α1, . . . , αd, βd+1, . . . , βn)∈Nⁿ. Then for anyC>0, there exists a bijectionbhinBox(Ch,Ch) betweenRes(P) and the set of pseudo-resonances such that bh(E) =E+O(h^3/2).

IThe cased=n(a barrier top) was also studied by Briet, Combes and Duclos under a virial condition. We denoteΓ(h)the set of theEα,β in that case.

IIn the cased= 0 (a well in an island), much more precise results have been proved. The imaginary part of resonances is exponentially small and the decay rate is given by the Agmon distance from the well to the sea.

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The geometrical setting Main result Some comments

The potential

IWe assume thatV has alocal non-degenerate maximum at 0

V(x) =E₀−

n

X

j=1

λ²_j

4x_j²+O(x³) with 0< λ₁≤ · · · ≤λ_n.

Brief Article

The Author November 6, 2012

V(x)

0

1

Brief Article

The Author November 6, 2012

{V(x) =E0}

0 2θ0 π(H)

1 IWe set

F_p=d_(0,0)H_p=

0 2Id

1

2diag(λ²₁,· · ·, λ²_n) 0

Notice thatσ(F_p) ={−λ_n,· · ·,−λ₁, λ₁,· · ·, λ_n}.

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Stable/unstable manifolds

IWe denoteΛ₋ andΛ₊the stableincomingandoutgoing manifoldat (0,0) respectively:

Λ_±=

(x, ξ); exp(tHp)(x, ξ)→0 as t→ ∓∞

IΛ₋ andΛ+are smooth, Lagrangian manifolds, stable under theHp

flow, given near (0,0) by

Λ_±=

(x, ξ); ξ=∇ϕ_±(x) with ϕ_±(x) =±

n

X

j=1

λj

4x_j²+O(x³)

IForρ_±∈Λ_±, one can see that

Πxexp(tHp)(ρ±) =g±(ρ±)e^±λ¹^t+O e^±(λ¹^+ε)t

ast→ ∓∞.

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Assumptions

(A1)The trapped set at energy E₀is

K(E₀) ={(0,0)} ∪ H, withH= Λ₋∩Λ₊\ {(0,0)} 6=∅.

His the set of the homoclinic curves.

(A2) ∀ρ₋, ρ+∈ H, g₋(ρ₋)·g+(ρ+)6= 0.

γ−(ρ−) (0,0)

H γ₊(ρ₊)

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Assumptions and notations

IWe focus here on the particular case where (A3)H=SJ

j=1γj, where γj are homoclinic trajectories

(A4)The intersection Λ+∩Λ− is transverse along each of theγj’s.

IForj ∈ {1, . . . ,J}, we denote:

I g_j^± the asymptotic directions of the trajectoryγj,

I A_j =R

γjξ·dx the classical action alongγ_j,

I νj the Maslov index alongγj.

IMorever, we set

E−E₀=hz, L:= 1 2

n

X

j=2

λ_j andζ=ζ(z) = 1 λ1

(L−iz).

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The main result

IConsider theJ×JmatrixM=M(ζ,h) = (mjk)j,k where

mjk(ζ) =Kjke^iA^j^/h Γ(ζ+¹₂) (iλ1g₊^j ·g₋^k)^ζ+¹²,

Kjk= r2π

λ1

e⁻^π²^(ν^j⁺¹²⁾ⁱ|g₋^k| lim

t→−∞

pDj(t) e⁽^λ²¹^+L)t

t→+∞lim

e⁽^λ²¹^−L)t pDk(t)·

Theorem (Bony, Fujiie, R., Zerzeri) Letµ1, . . . , µJ be the eigenvalues ofM, and

ΨRes(P) ={E ∈C;∃j∈ {1, . . . ,J}s.t.h^ζµj(ζ,h) = 1}.

For any small, we have, inB1() =Box(Ch,(Λ +λ1−)h)\{Γ(h) +D(h)},

dist(Res(P),ΨRes(P)) =o h

|logh|

.

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The case of 1 homoclinic cure

ILetσ∈R. The pseudo-resonancesE, such thatReE =E0+σh+o(h), are given by

E=E0+λ1

2kπ−A

h h

|logh|−i

Lh−λ1logm

−i σ λ1

h

|logh|

+o h

|logh|

, for somek∈Z.

2πλ1 h

|logh|

E0

E0−Ch E0+Ch

1 2

n

X

j=2

λjh O _|log^hh|

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Behavior of the resonances at the boundary of the box

IThe behaviour of the ”resonance curve” when (ReE−E0)/hgoes to

±∞can easily be seen. On that curve one has

ImE =−Lh+ h

|logh|







K−πReE−E₀

h +o(1) as ReE−E₀

h →+∞,

K+o(1) as ReE−E0

h → −∞, for some explicit constantK. Thus we see the transition from the

geometric situation for energiesE belowE0 (which is generally trapping) to the geometric situation aboveE0(which is generally non-trapping).

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Two homoclinic curves

ISupposeg_±¹·g_±² <0 (i.e. the homoclinic trajectories are on the opposite side).

– Whenσ <<−1,Mis almost diagonal, hence|µ1| ∼ |m11|,

|µ2| ∼ |m22|, and the ”resonance curve” consists of two lines on each of which the distance between two neighboring resonances is 2πh.

– Whenσ >>1,Mis almost anti-diagonal and

|µ1| ∼ |µ2| ∼p

|m12m21|, and the ”resonance curve” consists of a unique line in which the distance between two neighboring resonances isπh.

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The general case

Proposition

Letσ∈R. The pseudo-resonancesE, such thatReE =E0+σh+o(h), are given by

E =E₀+ 2kπλ₁ h

|logh| −iLh+iλ₁logµ_j h

|logh|+o h

|logh|

,

for somek ∈Z.

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Strategy

IOur results rely heavily on the study of what we call a microlocal Cauchy problem, of the form

((P−E)u=v microlocally in Ω, u=u− microlocally nearS−,

where Ω is some neighborhood of the trapped set, andS−is a suitable hypersurface in the incoming region of Ω.

I Our main resolvent estimate (and therefore the existence of a resonance free region) in some region of the complex plane of energies follows from a uniqueness result.

I The existence, and precise estimates on resonances follows from the existence of solution to the inhomogeneous microlocal Cauchy problem.

IWe obtain that way a Bohr-Sommerfeld like quantization condition for the resonances, which implies in particular that they accumulate on precisely defined curves ash→0.

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A resolvent estimate

IWe start by proving a polynomial estimate for the resolvent, away from the pseudo-resonances.

Proposition

∀C>0,∀ >0,∃M>0 such that if E∈B1()\

ΨRes(P) +D(_|log^h_h|)

,then, forθ=hlog(1/h), one has

(Pθ−E)⁻¹=O(h^−M).

IWe prove this theorem by contradiction. We assume that there exist >0,C >0,E ∈B1()\{ΨRes(P) +D(_|_log^h_h|)},u∈L²with||u||= 1 satisfying

(P_θ−E)u=O(h^∞), (1)

and deducekuk|=O(h^∞),which is a contradiction.

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Step 0

It is enough to show thatu= 0 microlocally on the trapped set.

Definition

Letρ∈R²ⁿ. We say thatu= 0microlocally nearρiff

Op(ψ)u

=O(h^∞) for someψ∈C₀^∞(R²ⁿ)withψ= 1nearρ.

ILocalization on the energy levelE0

SincePθ−z ≈P−z is elliptic outside of the energy levelE0, we have u=ϕ(P)u+O_H2(h^∞)

for allϕ∈C₀^∞(R) withϕ= 1 nearE0.

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ISpatial localization

Let (x, ξ) close to the energy levelE₀.

−Imp_θ(x, ξ)≈

(sin(2θ)ξ²≈2θE₀&θ for|x|>R O(θ)

forR1. Let χ∈C₀^∞(Rⁿ) withχ= 1 for|x|<R. This implies O(h^∞) =−Imh(Pθ−z)u,ui&θk(1−χ)uk²− O(θ)kχuk² Then

k(1−χ)uk.kχuk+O(h^∞)

ITherefore, it is enough to prove that

kϕ(P)χuk=O(h^∞)

By a compactness argument, it is enough to prove that u= 0 microlocally near ρ for allρ∈supp(ϕ(p(x, ξ))χ(x)).

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Step 1

ISupposeρ∈Λ−\Λ+. Lemma

exp(tHp)(ρ)→ ∞as t→ −∞ ⇐⇒ ρ /∈Λ+

ITherefore, there existsT>0 such that exp(−TH_p)(ρ)∈Γ⁻=n

(x, ξ)∈p⁻¹(E0);|x| 1,cos(x, ξ) = x·ξ

|x||ξ|<−1/2o

Theorem (Bony–Michel ’04)

u= 0microlocally near eachρ∈Γ⁻

IStandard propagation of singularities If for someT>>1,

((Pθ−z)u=O(h^∞)

u= 0 microlocally near exp(−TH_p)(ρ)

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Propagation through the critical point

Theorem (BFRZ’07)

Forε1, letS_ε={(x, ξ)∈Λ₋; |x|=ε}andρ₋∈S_εbe such that g₋(ρ₋)6= 0. Let u₋, withku₋k ≤1, be such that

((P−z)u₋= 0 microlocally near Sε

u₋= 0 microlocally near each point of S_ε\ {ρ₋}

Then, the problem

((P−z)u= 0 microlocally near(0,0) u=u₋ microlocally near S_ε

has a unique solution u withkuk ≤h^−C.

Moreover, ifρ₊∈Λ₊satisfiesg₋(ρ₋)·g₊(ρ₊)6= 0, then

u=h

Pλj−λ1 2λ1 −i_hλ^z

1

Z

e^i(ϕ⁺^(x)−ϕ⁻^(y^))/hd(x,y;h)u₋(y)dy microlocally nearρ₊.

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Step 2:

IOn Λ−,uis supported only on γ∪(0,0), and hence, by (A4), we can apply the previous Theorem for (x, ξ) near eachρ₊= (x₊, ξ₊)∈Λ₊∪γ.

In particular,uis of the form

u(x,h) =a₊(x,h)e^iφ⁺^(x)/h microlocally near ρ₊,

witha+(x,h)∈S(h^−c).

IThe standard Maslov theory tells us that, after continuation alongγ,u is also of the form

u(x,h) =a₋(x,h)eⁱ^φ^˜⁺^(x)/h microlocally near ρ₋ (2) witha−(x,h)∈S(h^−c) for somec∈R. Here ˜φ+is a generating

function of the evolution of Λ+nearρ−.

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Step 3

IStarting from (2), we continueuone more tour along (0,0)∪γusing propagation through the critical point and the Maslov theory. Then we obtain another expression ofumicrolocally near (x₋, ξ₋):

u(x,h) = ˜a₋(x,h)eⁱ^φ^˜⁺^(x)/h microlocally near ρ₋. (3)

Lemma

At the principal level, the value˜a₋(x,h)is determined by the value a(x₋,h)

˜

a₋(x,h) =M(x,x₋;ζ,h)a₋(x₋,h)(1 +O(h)), whereM(x,x₋;ζ,h)∈S(h^Re^ζ)is a symbol satisfying

M(x₋,x₋;ζ,h) =h^ζm(ζ)e^iA/h.

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IWe have, for each pointρ⁰= (x⁰, ξ⁰)∈Λ+ withg+(x⁰)·g₋(x₋)6= 0,

u(x⁰,h) =a+(x⁰,h)e^iφ⁺^(x)/h, a+(x⁰,h) =J(x⁰,x−;ζ,h)a−(x−,h). (4)

IOn the other hand, by Maslov theory, ˜a(x,h) can be expressed by the valuea+(x,h) whenρ= (x, ξ)∈eΛ+is connected by a classical trajectory withρ⁰= (x⁰, ξ⁰):

˜

a₋(x,h) =e^iA(ρ⁰^,ρ)/he⁻^π²^νiI_ρ⁰_→ρa₊(x⁰,h). (5)

HereI_ρ⁰_→ρ is a multiplication operator given by

Iρ⁰→ρ= s

D(tρ⁰,y⁰) D(tρ,y⁰)|_y⁰_=y⁰

0,

whereρ= (x(tρ,y₀⁰), ξ(tρ,y₀⁰)) andρ⁰= (x(tρ⁰,y₀⁰), ξ(tρ⁰,y₀⁰)).

Combining (4) and (5), we obtain the lemma. Note thatM(x₋,x₋) is independent of the choice ofx₋.

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Step 4

We have obtained that, microlocally nearρ−,

a₋(x,h) =M(x,x₋;ζ,h)a₋(x₋,h) +S(h^Re^ζ−c+1) (6) Atx =x₋, one has

a−(x−,h) = O(h^Re^ζ−c+1)

1− M(x₋,x₋) = O(h^−c+1) h^−ζ−me^iA/h

Lemma

E∈/ ΨRes(P) +D(_|_log^h_h|)implies|h^−ζ−me^iA/h| ≥ν for an h-independent positive constantν.

Hence

a₋(x₋,h) =O(h^−c+1).

Substituting this into (6), one gets

a₋(x,h)∈S(h^Re^ζ−c+1).

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IEventually,ImE>(−L−λ1+)himpliesReζ >−1 + _λ

1. Hence a₋(x,h)∈S(h^−c+^λ¹).

Thus we obtaina− ∈S(h^−c+^λ¹) from the assumptiona−∈S(h^−c). We conclude by bootstrap that

a−(x,h)∈S(h^∞),

which is the contradiction we were looking for.

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Existence of resonances

IWe prove the existence of a resonance close to each pseudo-resonance also by a contradiction argument. We assume that there existε >0 and Ee∈ΨRes(P)∩B1(C, ε) such that there is no resonance in the interior of the circleC:={z ∈C; |z−Ee|= _|_log^εh_h|}.

IForE ∈ C, we consider the inhomogeneous equation

(Pθ−E)u=v, (7)

wherev is suitably chosen.

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Construction of v

ILetH₋ be a hyperplane containingx =x₋ and transversal to Π_xγ, and let ˜v(x) = ˜χ(x,h)eⁱ^φ^˜⁺^(x)/h be the WKB solution of the microlocal Cauchy problem

((P−E)˜v = 0,

˜

v=χ(x)eⁱ^φ^˜⁺^(x)/h onH₋, whereχ∈C₀^∞andχ= 1 nearx−.

IWe set

v(x) =ψ[P, ϕ]˜v,

whereψandϕare smooth cutoff functions with small compact support such thatϕis equal to 1 in a neighborhood ofx₋, and that ψis equal to 1 onsupp[P, ϕ]˜v∩γ₊, whereγ₊=S

t<0exptH_p(ρ₋). Obviously,v is holomorphic inE near E₀.

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IThe solutionu= (Pθ−E)⁻¹v of (7) is well-defined and satisfies kuk=O(h^−c). Moreover, working as in (2), one can prove thatu is of the form

u=a−(x,h)eⁱ^φ^˜⁺^(x)/h, a−∈S(h^−c), (8) microlocally nearx−.

IThanks to (7), (P−z)u= 0 microlocally near (0,0). Moreover, on Λ₋,uis supported microlocally onγ. Thusu is of the form

u₊(x⁰,h) =a₊(x⁰,h)e^iφ⁺^(x)/h,

with

a+(x⁰,h) =J(x⁰,x₋;ζ,h)a₋(x₋,h) +S(h^Re^{ζ−c+1−µ}), (9) microlocally on Λ+∩γ.

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IOn the other hand, alongγ,usatisfies the following microlocal Cauchy problem:

((P−E)u=v microlocally along γ, u=u₊ microlocally near S₊^ε ∩γ.

The solution of this linear problem can be written as the sumu1+u2 of the solutions of the microlocal Cauchy problems

((P−E)u1= 0 alongγ,

u1=u+ nearS₊^ε∩γ, and

((P−E)u2=v alongγ, u2= 0 nearS₊^ε ∩γ.

As in (5), we have

u1(x,h) =

e^iA/he⁻^π²^νie^izTIρ⁰→ρa+(x⁰,h) +S(h^Re^ζ−c+1)

eⁱ^φ^˜⁺^(x)/h. Moreover,v has been constructed so thatu₂=ϕ˜v microlocally nearρ₋.

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ISumming up, we have obtained

a₋(x,h) =e^iA/he⁻^π²^νie^izTIρ⁰→ρa+(x⁰,h) + ˜χ(x,h) +S(h^Re^{ζ−c+1−µ}), (10) nearx₋. Combining (9) and (10), we obtain

a₋(x,h) =M(x,x₋;ζ,h)a₋(x₋,h) + ˜χ(x,h) +S(h^Re^{ζ−c+1−µ}). (11)

and we can show thata−∈S(1).

IThus (11) becomes

a−(x) = h^−ζ

h^−ζ −me^iA/hχ(x,˜ h) +S(h^1−µ).

We integrate this identity with respect toE along the contourC, and we obtain

Z

C

a₋(x)dE= 2πλ1

h

loghχ+S(h^2−µ).

This is a contradiction because the left hand side should vanish if there is no resonance inside the contourC.