Résonances et états résonants pour des ensembles captés singuliers

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Resonances and resonant states for singular trapped sets

Thierry Ramond

Laboratoire de Math´ematiques d’Orsay, Univ. Paris Sud, CNRS, Universit´e Paris Saclay

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

Nice, 2015/11/05

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We are interested in quantum resonances in the presence of classical orbits that are singular. We consider the case where the Hamiltonian vector field has singular points of hyperbolic type, and some associated homoclinic or heteroclinic orbits.

We have proved

I semiclassical resolvant estimates, which imply the existence of some resonance free zone below the real axis of energies,

I the existence of resonances,

I that these resonances are given by a Bohr-Sommerfeld like

quantization rule. In particular they concentrate on curves ash→0.

I a precise localisation of these resonances.

I some knowledge on the microlocal localization of the corresponding resonant states

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Example: the case of 1 homoclinic curve

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

E=E0+λ1

2kπ−A

h h

|logh|−i

Lh−λ1logm

−i τ 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|

IExistence of resonances:

dist^∗ Res(P),ΨRes(P)

=o h

|logh|

,

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Outline of the talk

Introduction

Resonances for the Schr¨odinger operator The trapped set

The Homoclinic case The geometrical settings Assumptions and Notations Main results

Resonant States

The multi-bump case Settings

Pseudoresonances Resonances

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Plan of the Lecture

Introduction

Resonant States

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Eigenvalues vs resonances,

Classical vs Quantum mechanics

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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Schr¨ odinger operators

IWe consider the Schr¨odinger operator onRⁿ,n≥1, P=−h²∆ +V(x) =Op^w_h(p) = 1

(2πh)ⁿ ZZ

e^i(x−y^)·ξ/hpx+y 2 , ξ

dy dξ

wherep(x, ξ) =ξ²+V(x),V(x)∈C^∞(Rⁿ;R), andV(x) has an analytic extension to

Σ ={x ∈Cⁿ; |Imx|< δ|Rex| and|Rex|>C}

for someδ,C>0. We suppose that V(x)−→0 as |x| → ∞,x ∈Σ.

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 and resonances

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→Rⁿ be 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 ∇(xe^µF(x))^1/2ϕ xe^µF(x)

IForθ∈Rsmall enough, we denotePθ=UiθPU_i⁻¹_θ the distorted operator. The set of resonances ofPinE^θ={−2θ <ImE<0}is

Resθ(P) =σ(Pθ)∩ E^θ

Cⁿ

xe^iθF(x) Σ

Res(P) Eθ C

−2θ

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The trapped set

Definition:

The trapped set at energyE0>0 is K(E₀) =

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

Hp is the Hamiltonian vector field onT^∗Rⁿ=R²ⁿ associated top, given byHp=∂ξp∂x−∂xp∂ξ =

2ξ

−∇V(x)

.

IKnown results:

- An empty trapped set (Helffer-Sj¨ostrand, Martinez).

- Le puits dans l’isle (shape resonances) (Helffer-Sjöstrand) - A hyperbolic closed trajectory (C.Gérard - Sjöstrand)

- A non-degenerate critical point (Briet-Combes-Duclos, Sj¨ostrand) - 1-dimensional situations.

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Plan of the Lecture

Introduction

Resonant States

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The Homoclinic case

V(x)

0

{V(x) =E0} π(H) 0

1

IWe assume thatV has alocal non-degenerate maximum at 0

V(x) =E₀− Xn

j=1

λ²_j

4x_j²+O(x³)

with 0< λ1≤ · · · ≤λn.

IWe set

F_p=d_(0,0)H_p=

0 2Id

1

2diag(λ²₁,· · ·, λ²_n) 0

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

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Stable/unstable manifolds

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

Λ_±=

(x, ξ); exp(tH_p)(x, ξ)→0 as t→ ∓∞

IΛ₋ andΛ₊are smooth, Lagrangian manifolds, stable under theH_p flow, given near (0,0) by

Λ_±=

(x, ξ); ξ=∇ϕ±(x) with ϕ±(x) =± Xn

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.

IIn this section we consider the case where (A3)H=S^K

k=1γk, where γk are homoclinic trajectories (A4)The intersection Λ+∩Λ₋ is clean along each of theγk’s:

∀ρ∈ H TρΛ−∩TρΛ+=Tρ(Λ−∩Λ+) =RHp

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Notations

IFork ∈ {1, . . . ,K}, we denote:

I g_k^± the asymptotic directions of the trajectoryγk,

I Ak =R

γjξ·dx the classical action alongγk,

I νk the Maslov index alongγk.

IWe set

L:= 1 2

Xn

j=2

λj andζ=ζ(z,h) = 1 λ1

(L−iz−E₀ h ).

IConsider theK ×K matrixQ=Q(ζ,h) = (q`,k)`,k where

q`k(ζ) = e^iA^`^/h

(iλ1g₊^j ·g₋^k)^ζ+¹²Kjk,

Kjk=√2π

λ1Γ(ζ+ 12)e−π

2(νj+ 12)i|g k− |limt→−∞

√Dj(t) e(λ1

2 +L)t

limt→+∞e(λ1 2−L)t

√Dk(t) ·

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Quantization rule and pseudo-resonances

Letµ1, . . . , µJ be the eigenvalues ofQ, and

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

IThis is a quantization rule: z is a pseudo-resonance if and only if 1∈σ h^ζQ(z,h)

Theorem (BFRZ)

Forτ∈R, the pseudo-resonancesz such that Rez =E0+τh+o(h) are given byz =z_q,k(τ) +o

h

|lnh|

, where

z_q,k(τ) =E₀+ 2qπλ1

h

|lnh|−ih Xn

j=2

λj

2 +iλ1ln(µk(τ,h)) h

|lnh| for someq∈Zandk ∈ {1, . . . ,J}.

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Existence and asymptotics of the resonances

Theorem (BFRZ)

LetC, δ >0. In the domain

E0+ [−Ch,Ch] +ih

−hL−C h

|lnh|,0i

\ Γ(h) +B(0, δh) ,

we havedist^∗ Res(P),ΨRes(P)

=o

h

|lnh|

,ash→0. Moreover, for allχ∈C₀^∞(Rⁿ), there existsM >0 such that

χ(P−z)⁻¹χ.h^−M,

uniformly forhsmall enough andz such that dist(z,ΨRes(P))≥δ_|_ln^h_h|.

Definition

Let A,B,C be subsets ofCandε≥0. We say that dist^∗(A,B)≤εin C, if and only if

∀a∈A∩C ∃b∈B |a−b| ≤ε, and ∀b∈B∩C ∃a∈A |a−b| ≤ε.

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Existence and asymptotics of the resonances

Theorem (BFRZ)

E0+ [−Ch,Ch] +ih

−hL−C h

|lnh|,0i

\ Γ(h) +B(0, δh) ,

we havedist^∗ Res(P),ΨRes(P)

=o

h

|lnh|

,ash→0. Moreover, for allχ∈C₀^∞(Rⁿ), there existsM >0 such that

χ(P−z)⁻¹χ.h^−M,

uniformly forhsmall enough andz such that dist(z,ΨRes(P))≥δ_|_ln^h_h|.

Definition

Let A,B,C be subsets ofCandε≥0. We say that dist^∗(A,B)≤εin C, if and only if

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Two scale asymptotics for the resonances

16 J.-F. BONY, S. FUJII´E, T. RAMOND, AND M. ZERZERI

C_|_ln^h_h_|

E0

E0−Ch E0+Ch

1 2

!n j=2

λjh 2πλ1 h

|lnh|

Figure 7. Two scale asymptotic of resonances of Theorem4.6.^d8 f23

withRez∈E0+τ h+hδ(h)[−1,1]satisfy d94

d94 (4.10) z=z_q,k(τ) +o" h

|lnh|

# , with

d95

d95 (4.11) z_q,k(τ) =E₀+ 2qπλ₁ h

|lnh|−ih

!n j=2

λj

2 +iln(µ_k(τ, h))λ₁ h

|lnh|,

for someq ∈ Z and k ∈ {1, . . . , K}. On the other hand, for each τ ∈ [−C, C], q ∈ Z and k∈ {1, . . . , K}such thatzq,k(τ)belongs to(4.9)^d12 with a real part lying inE0+τ h+hδ(h)[−1,1], there exists a pseudo-resonancez satisfying (4.10)^d94 uniformly with respect toq, k, τ.

We do not need to specify the determination of the logarithm ofµ_k(τ, h) in (^d954.11). Indeed, a change of determination is balanced by a change ofq∈Z. Note also thatzq,k(τ) is in (4.9)^d12 only for eigenvaluesµk(τ, h) outside a vicinity of 0.

To compare the set of the resonances and the set of the pseudo-resonances, we will use the following definition. Its flexibility avoids the problems which may occur at the boundary of the domain of study.

g80 Definition 4.5. LetA, B, C be subsets ofCandε≥0. We say that dist(A, B)≤εin C,

if and only if

For a finite number of homoclinic trajectories, our main result is the following.

d8 Theorem 4.6(Asymptotic of resonances). Assume (H1)–^h1 (H4),^h4 (H7)^h8 and let C, δ > 0. In the domain

d90

d90 (4.12) E0+ [−Ch, Ch] +i$

−

!n

j=2

λj

2 h−C h

|lnh|,0%

\&

Γ(h) +B(0, δh)' ,

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About the multiplicity

Proposition

Lets>0 be small enough. Then, for any pseudo-resonance z, we have

card

Res(P)∩B

z,2s h

|lnh|

≥cardn

(q,k)∈Z× {1, . . . ,K}; zq,k(τ)∈B z,s h

|lnh| o,

whereτ= (Re(z)−E₀)/h. In the previous expression, the resonances are counted with their multiplicity.

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Example: the case of 1 homoclinic curve

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

E=E0+λ1

2kπ−A h

h

|logh|−i

Lh−λ1logm

−i τ 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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Example: Two homoclinic curves

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

– Whenτ <<−1,Qis almost diagonal, hence|µ1| ∼ |q11|,|µ2| ∼ |q22|, and the ”resonance curve” consists of two lines on each of which the distance between two neighboring resonances is 2πh.

– Whenτ >>1,Qis almost anti-diagonal and|µ1| ∼ |µ2| ∼p

|q₁₂q₂₁|, and the ”resonance curve” consists of a unique line in which the distance between two neighboring resonances isπh.

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Example: Three homoclinic curves

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Vibration of the accumulation curves

Let us consider how accumulation curves depends onh.

Imσ=− Xn

j=2

λj

2 + ln |µk(Reσ,h)| λ1

|lnh| where theµk(τ,h) are the eigenvalues of the matrix

Q= (mjk), qjk(ζ) = e^iA^j^/h

(iλ1g₊^j ·g₋^k)^ζ+¹²Kjk,

which is quasi-periodic with respect toh⁻¹.

Vibration

– if all the actions are equal, the accumulation curves do not dependent onh.

– if exactly two actions are different, sayA1andA2, the accumulation curves are periodic with respect toh⁻¹, with period 2π|A₂−A₁|⁻¹. – else accumulation curves are no more periodic w.r.t. h⁻¹but they are continuous functions ofe^i(A²^−A¹^)/h, . . . ,e^i(A^K^−A¹^)/h. This is what we call the vibration phenomenon.

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Resonant States

Theorem(BFRZ)

Letv=v(h) be a normalized resonant state associated to a resonance z=z(h). Then there ish^M ≤c(h)≤h^−M, such that, ifu=cv,

I MS(u)⊂ {(0,0)} ∪Λ₊.

I The function uis inI(Λ+,1) microlocally near any point ofH. We write

u(x,h) =e^−iA^k^/heⁱ^z−E^h⁰^t⁻^k M⁻k

D^k(t₋^k)a^k₋(x,h)e^iϕ¹⁺^(x)/h, microlocally near ρ^k₋, for somea^k₋∈S(1).

I Let A0(h) = (A0¹(h), . . . ,A0^K(h))∈C^K be defined by A0^k(h) =a^k₋(x₋^k,h). ThisK-vector satisfies the equation

h^S(z,h)/λ¹^−1/2Q(z,h)−1A0(h) =O(h¹²), ashgoes to 0, with the normalizationkA0(h)k= 1.

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Plan of the Lecture

Introduction

Resonant States

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The multi-bump case

ISettings

−→ The Hamiltonian vector fieldHp has a finite number of non-degenerate hyperbolic fixed points in the energy surfacep⁻¹(E₀).

−→ The trapped setK(E₀) at energyE₀ consists of these points (we call them vertices) and a finite number of homoclinic or heteroclinic curves (we call them edges)

−→ These edges are the transversal intersections of an outgoing stable manifold associated with a fixed point and an incoming stable manifold associated with a fixed point (the same or not).

ITerminology

−→ Every edgee is directed as an integral curve forHp. It starts at a vertex, and it ends at a vertex.

−→ We say that an edgee is successive to another edgee⁰and writee⁰→e if the endpoint ofe⁰is the starting point of e.

−→ A pathγ is a finite, successive sequence of edges inV.

−→ If the first edge of a pathγ is successive to the last edge, we say thatγ is a cycle.

−→ A cycle is called primitive if it does not contain any non-trivial sub-cycle.

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Damping associated to a cycle

I For each vertexv ∈ V, we denoteλ^v₁≤λ^v₂≤ · · · ≤λ^v_n thenpositive eigenvalues ofdHp(v), and we set

A_v =1 2

Xn

k=2

λ^v_k

λ^v₁, B_v = 1 λ^v₁·

I For a cycleγ, we denoteV(γ) the set of the vertices that belong to an edge inγ. Then we set

A(γ) = X

v∈V(γ)

Av, B(γ) = X

v∈V(γ)

Bv,

I Finally, we define the damping for the cycleγ to be

D(γ) = A(γ) B(γ)·

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Why?

Theorem (BFRZ’07)

Forε1, letSε={(x, ξ)∈Λ−; |x|=ε}andρ−∈Sε be 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 Sε\ {ρ−} Then, the problem

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

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

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

P^λj−λ1 2λ1 −i_hλ^z

1

Z

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

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Definition of the pseudoresonances

IWe suppose that there is only one primitive cycleγ⁰such that D⁰:=D(γ⁰) = min

γ D(γ), where the minimum is taken over all (primitive) cycles.

IWe denote ΨRes(P) the set of complex numbersz such that h^A(γ⁰^)−iB(γ⁰^)(z−E⁰^)/h= 1.

ILete be any edge in the minimal cycleγ⁰with starting pointv. There exists unique edgee⁰ inγ⁰such thate⁰→e. We set

Jê←e⁰(z,h) :=eîAê^/hΓ S_v

λ^v₁ rλ^v₁

2π M⁺e

M⁻ê e⁻^π²^(νê⁺¹²⁾ⁱg₋ê⁰ iλ^v1g₊ê·g₋ê⁰−Sv/λ^v₁

, whereA_e:=R

eξ·dx is the action along e,S_v =Pn j=1

λ^v_j

2 −i(z−E₀)/h, and

Q(z,h) = Y

e∈γ⁰

J^e←e⁰(z,h).

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Asymptotics of the pseudoresonances

Theorem (BFRZ)

LetC>0 and letδ(h) be a function which goes to 0 ash→0. Then, uniformly forτ ∈[−C,C], the pseudo-resonancesz in

E₀+ [−Ch,Ch] +ih

− D⁰h−C h

|lnh|,0i ,

withRez∈E0+τh+hδ(h)[−1,1] satisfyz =zq(τ) +o

h

|lnh|

,with

zq(τ) =E0+ 2qπλ1

h

|lnh|−ih Xn

j=2

λj

2 +iln(µ(τ,h))λ1

h

|lnh|, for someq∈Z, where µ(τ,h) :=Q(E0+τh−iD⁰h,h).

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Resonances

Theorem (BFRZ)

E₀+ [−Ch,Ch] +ih

− D⁰h−C h

|lnh|,0i

\ Γ(h) +B(0, δh) ,

we have

dist^∗ Res(P),ΨRes(P)

=o h

|lnh| ,

ashgoes to 0. Moreover, for allχ∈C₀^∞(Rⁿ), there exists M>0 such

that χ(P−z)⁻¹χ.h^−M,

uniformly forhsmall enough andz such that dist(z,Res0(P))≥δ_|_ln^h_h|·