Quantum resonances: a brief survey

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Quantum resonances:

a brief survey

Introduction Mathematical definitions Resonances in 1d Some examples The trapped set Semiclassical resolvent estimates An application : the quantum evolution

Quantum resonances in the semiclassical regime: a brief survey

Thierry Ramond

Département de Mathématiques, Université Paris Sud, UMR CNRS 8628.

Ritsumeikan University, 21-23/10/2010

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Quantum resonances:

a brief survey

Plan of the Lecture

Introduction

Breit-Wigner peaks The Schr¨odinger operator Mathematical definitions

Poles of the resolvent Poles of the scattering amplitude

Classical scattering Quantum scattering

Eigenvalues of a deformed operator

More general deformations Resonances in 1d

Some examples Le puits dans l’isle Barrier-top resonances Homoclinic resonances

The trapped set Settings

Definition and examples When the trapped set is empty

Semiclassical resolvent estimates A universal upper bound The barrier top case An application : the quantum evolution

The shape resonances case The barrier top case

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Quantum resonances:

a brief survey

Introduction Breit-Wigner peaks The Schr¨odinger operator Mathematical definitions Resonances in 1d Some examples The trapped set Semiclassical resolvent estimates An application : the quantum evolution

Introduction

I In 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

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Quantum resonances:

a brief survey

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

w_a,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.

I Such complex values for energies had also appeared for example in the work by Gamow, to explainα-radioactivity, and the notion of resonance was therefore also associated to some energy decay.

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Quantum resonances:

a brief survey

The Schr¨ odinger operator

I A quantum particle of massm, moving in a potential

V ∈ C^∞(Rⁿ,R), is described by a solution of the time-dependent Schr¨odinger equation

i~∂tφ(t,x) =P(x,~Dx)φ(t,x), P(x,~D_x) =−_2m^~²∆ +V(x).

I WhenE ∈σ_disc(P), andPφ₀=Eφ₀,kφ0kL²= 1, then φ(t,x) =e^−iEt/^~φ₀(x,~)

is a solution, which satisfies the normalization condition kφ(t, .)k_L2 =Z

|φ(t,x)|²dx^1/2

= 1.

φ(t,x) is called the wave function of the particle, with energyE, and|φ(t,x)|²is its density of probability of presence at timet.

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Quantum resonances:

a brief survey

INow supposeψ₀is a resonant state (not inL²) corresponding to the resonanceρ=a−ib/2. Its time evolution should be written

ψ(t) =e^{−ita−tb/2}ψ₀,

so that its density of probability of presence at time t is

|ψ(t,x)|²

|ψ0(x)|² =e^−bt, andbis the decay rate of that probability.

I In other words : a resonanceρ∈Ccorresponds to a

quasi-particle with lifetime 1/Imρ. The more a resonance is close to the real axis, the more its presence leads to physically

observable effects (can also be seen on Breit-Wigner peaks).

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Quantum resonances:

a brief survey

Introduction Mathematical definitions

Classical scattering Quantum scattering Eigenvalues of a deformed operator More general deformations Resonances in 1d Some examples The trapped set Semiclassical resolvent estimates An application : the quantum evolution

Poles of the resolvent

I We suppose thatV ∈ C^∞(Rⁿ) is real valued, and that

∂^αV(x) =O(hxi^−σ−|α|), for some σ >0.

I ThenP=−h²∆ +V is self-adjoint with domainH²(Rⁿ). In particular forImz >0,R(z) = (P−z)⁻¹:L²(Rⁿ)→L²(Rⁿ) is well defined.

Theorem and Definition 1 :

The resolvent

{Rez>0,Imz >0} 3z 7→R(z) :L²_comp(Rⁿ)→L²_loc(Rⁿ) can be meromorphically continued to the lower half-plane {Rez >0,Imz <0}. Its poles are called resonances ofP.

0 Im z

Re z σess(P)

σdisc(P)

(P-z)-1

R(z)

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Quantum resonances:

a brief survey

Classical scattering

I A classical particle with mass 1, when placed in the conservative force fieldF(x) =−∇V(x), follows the trajectory

(x(t,x₀, ξ₀), ξ(t,x₀, ξ₀)) given by x˙(t) =ξ(t),

ξ(t˙ ) =−∇V(x(t)).

I This is Newton’s Equation written in Hamiltonian form : In the phase spaceT^∗R^d, the trajectory is an integral curve of the vector field

Hp=∂ξp∂x −∂xp∂ξ

associated to the total energy of the particlep(x, ξ) =¹₂ξ²+V(x)

I Note that :

Pu(x) =Op^w_h(p(x, ξ))u(x) = Z

e^{i(x−y)·ξ/h}p(x+y

2 , ξ)u(y)dy dξ (2πh)ⁿ·

I Scattering :V →0 at∞, and we consider trajectories going to infinity. The aim could be to recoverV knowing what goes out of the interaction region when one throw particles towards that region.

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Quantum resonances:

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I LetE >0 be fixed. For anyα∈Sⁿ⁻¹,z ∈α^⊥∼Rⁿ⁻¹, there is one and only one integral curve for Hp

γ_±(t,z, α,E) = (x_±(t,z, α,E), ξ_±(t,z, α,E))∈p⁻¹(E) such that

lim_t→±∞|x_±(t,z, α)−√

2Eαt−z|= 0, lim_t→±∞|ξ_±(t,z, α)−√

2Eα|= 0.

!

! z

"

z’

I The classical scattering map at energyE is the map (z, ω)7→(z⁰, θ).

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Quantum resonances:

a brief survey

The scattering amplitude

IWe suppose for simplicity thatV is a very short range potential :

∂^αV(x) =O(hxi^−σ−|α|), for someσ >(n+ 1)/2.

I LetE >0 be fixed. For anyω, θ6=ω∈Sⁿ⁻¹, there exists one and only oneu∈L²_loc(Rⁿ) such that

( Pu=Eu, u(x,h) =eⁱ

√E x·ω/h+A(ω, θ,E,h)^eⁱ

√E x·θ/h

|x|^(n−1)/2 +o(|x|^(1−n)/2), as|x| →+∞with_|x|^x =θ.

I The scattering amplitude at energyE is the operator A(E) :L²(Sⁿ⁻¹)→L²(Sⁿ⁻¹) with kernelA(ω, θ,E,h). The scattering matrix at energyE is

S(E) =I −2iπc(E,h)A(E), for some normalization constantc(E,h).

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Quantum resonances:

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More general potentials

I We suppose now thatV is a short range potential :

∂^αV(x) =O(hxi^−σ−|α|), for some σ >1, and we setP=P0+V,P0=−h²∆.

I For any givenu₋, one can findu, and thenu+, such that ke^−itP⁰^/hu₋−e^−itP/huk →0 ast → −∞, ke^−itP⁰^/hu+−e^−itP/huk →0 as t→+∞.

e-itP⁰ u_- e-itP⁰ u₊ e-itP/hu

/h

u- /h u₊ S u=W₊u₊

=W-u-

I The Wave operatorsW_±are defined byu=W₋u₋=W₊u₊.

I Notice the intertwining property :PW_±=W_±P₀.

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Quantum resonances:

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The scattering amplitude

I The scattering operator for (P,P₀) is defined by S = (W+)⁻¹W₋: u₋ 7→u+.

I SinceP0S − SP0= 0, one has S =

Z +∞

0

S(E)dµP(E), S(E) :L²(Sⁿ⁻¹)→L²(Sⁿ⁻¹), The operatorS(E) is the scattering matrix at energyE.

I T(E) = _2iπ¹ (Id− S(E)) is an integral operator, with a smooth kernel out ofθ=ω. Up to a multiplicative constant,T(E) is the scattering amplitude at energy E.

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Quantum resonances:

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Meromorphic extension of the scattering amplitude

Theorem and Definition 2 :

SupposeV is a long range potential. The scattering amplitude R^∗+3E 7→ A(E) can be meromorphically continued to the lower half-plane{Rez >0,Imz <0}. Its poles are called resonances of P.

I When isV compactly supported, one can prove that for χ₁≺χ₂∈ C₀^∞(Rⁿ),

A(ω, θ,E,h) =iπEⁿ⁻²² h(P−(E+i0))⁻¹[h²∆, χ2]eⁱ

√E x·ω/h, [h²∆, χ1]eⁱ

√E x·θ/hi

Then one can extendA(ω, θ,E,h) to complex values ofE by A(ω, θ,z,h) =iπzⁿ⁻²² hR(z)[h²∆, χ2]eⁱ

√zx·ω/h,[h²∆, χ1]eⁱ

√zx·θ/hi

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Quantum resonances:

a brief survey

Analytic dilation

I Forµ∈R, we set

U_µ :L²(R^d)→L²(R^d),U_µφ(x) =e^dµ/2φ(xe^µ), forφ∈ C₀^∞(R^d) and we see

UµPU_µ^∗=e^−2µ(−h²∆) +V(xe^µ).

Definition

The funtion V :R^d→Ris dilation-analytic when

µ7→V(xe^µ)(−h²∆ + 1)⁻¹extends as an analytic (w.r.t.µ) family of compact operators on L²(R^d).

I For example, ifV ∈ C^∞(R^d) extends as holomorphic function in Ω_θ₀ ={x∈C^d,|Imx| ≤tanθ₀hRexi},

where it satisfies

V(x)→0 as|x| →+∞, then V is dilation-analytic.

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Quantum resonances:

a brief survey

Resonances

I For such aV,

UµPU_µ^∗=e^−2µ(−h²∆) + [V(xe^µ)(−h²∆ + 1)⁻¹](−h²∆ + 1), so that one can definePθ=UiθPU_iθ^∗ forθ∈R⁺small enough.

I By Weyl’s Theorem

σ_ess(P_θ) =e^−2iθσ_ess(−h²∆) =e^−2iθR⁺.

Definition 3

Let V be analytic-dilation in|µ|< δ. The setRes(P)of resonances of P is

Res(P,h) = [

0<θ<δ

σdisc(Pθ)∩(0,R⁺,e^−2iθR⁺).

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Quantum resonances:

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Complex distortion

I Now we assume only thatV(x) has an analytic extension to Σ ={x∈Cⁿ; |Imx|< δ|Rex|and|Rex|>C}

for some δ,C>0, and thatV(x)→0 as |x| → ∞, x∈Σ.

ILetF :Rⁿ→Rⁿbe s. t.F(x) = 0 for|x|<C andF(x) = 1 for

|x| 1. Forµ∈Rsmall enough, we define Uµ:L²(Rⁿ)→L²(Rⁿ) as the unitary operator

Uµϕ(x) =

det d xe^µF^(x)

1/2ϕ xe^µF^(x) For θ∈Rsmall enough, we denote

P_θ=U_iθPU_iθ⁻¹

the distorted operator. As before, the set of resonances of Pis Res(P,h) = [

0<θ<δ

σdisc(Pθ)∩(0,R⁺,e^−2iθR⁺).

Cⁿ

xe^iθF(x) Σ

Res(P) Γ_θ C

−2θ

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Quantum resonances:

a brief survey

All these definitons coincide !

Theorem : (Helffer-Martinez)

When their domain of validity overlap, these different definitions of resonance (as well as more sophisticated ones) co¨ıncide.

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Quantum resonances:

a brief survey

Introduction Mathematical definitions Resonances in 1d

Some examples The trapped set Semiclassical resolvent estimates An application : the quantum evolution

Scattering in 1d

I The (stationary) Schr¨odinger equation in 1d is

−h²u⁰⁰+Vu=Eu.

I The scattering matrixS(E,h) gives outgoing data from incoming data : for V ∈ C₀^∞(R,R), and withk =√

E, for any solutionu(not inL²!) , one has

u1x<−R =p−e^ikx/h+q−e^−ikx/h, u1x>R =p+e^−ikx/h+q+e^ikx/h,

e+ikx/h

e+ikx/h e-ikx/h e-ikx/h

Then S(E) is defined by q+

q₋

=S(E,h) p+

p₋

.

I On can show that S(E,h) = 1

a^∗(E,h)

1 −b^∗(E,h) b(E,h) 1

where a(E) and b(E) are analytic. The resonances are the zeros ofa^∗(E).

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Quantum resonances:

a brief survey

Introduction Mathematical definitions Resonances in 1d

Some examples The trapped set Semiclassical resolvent estimates An application : the quantum evolution

I Now we supposeP=h²D²+V(x) withV holomorphic in {x∈C,|Imx|<tanθ0|Rex|}, and thatV is short range in that set. We have

Pθu(x) =UθPU−θu(x) = (−h²u⁰⁰(x) +V(xe^iθ)u(x).

I We denote byJ_±^d,g the solutions of (P−E)u= 0 characterized by

( J_±^r(x,E,h)∼e^±i

√E x/h as Rex→+∞, J_±^l (x,E,h)∼e^±i

√

E x/h as Rex→ −∞.

Then, for realE’s,J_±,θ^r,l :x7→J_±^r,l(xe^iθ) form a basis of solutions toPθu=EuinL²(R⁺) andL²(R⁻) respectively .

I ForE ∈Cwith−2θ0<argE <0,θ < θ₀ the only solution in L²(R⁺) isJ_+,θ^r (x,E,h), andJ_−,θ^l (x,E,h) is the only one in L²(R⁻).

I E∈Res(P) iff there is an outgoing solution, i.e.a^∗(E,h) = 0.

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Quantum resonances:

a brief survey

Introduction Mathematical definitions Resonances in 1d Some examples

Le puits dans l’isle Barrier-top resonances Homoclinic resonances The trapped set Semiclassical resolvent estimates An application : the quantum evolution

Le puits dans l’isle

I We suppose now thatV ∈ C^∞(Rⁿ,R) has an analytic extension to

Σ ={x∈Cⁿ; |Imx|< δhRexi}

for some δ >0, and thatV(x) =O(hxi^−σ),σ >0, as|x| → ∞, x ∈Σ. We also assume thatV has a non-degenerate local minimum atx0, and ”looks like”

E0

I LetK be a compact set containing a vicinity ofx0. We denote P^D the Dirichlet realization ofP inK. The spectrum ofP^D is discrete and we denote its eigenvalues byE1(h),E2(h), . . ..

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Quantum resonances:

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Shape resonances

Theorem (Helffer-Sj¨ ostrand)

There exists a bijectionb(h) between the spectrum of P^D in D(E0,Ch) and the set of resonances ofP inD(E0,Ch). Moreover

|b(z)−z|=O(e^−(2S⁰^/h)),

where S0is the Agmon distance betweenx0and the sea.

E0 E1(h) E2(h) Ek(h)

e-2S0/h

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Quantum resonances:

a brief survey

Barrier-top resonances

I We suppose thatV has an analytic extension to Σ ={x∈Cⁿ; |Imx|< δhRexi}

for some δ >0, and thatV(x)−→0 as |x| → ∞, x∈Σ.

Moreover, we assume

V(x) =E0−

n

X

j=1

λ²_jx_j²+O(|x|³).

and that the trapped set at energyE₀is{0}.

Theorem (Sj¨ ostrand, Briet-Combes-Duclos)

Let C >0be different fromPn

j=1(αj+¹₂)λj for allα∈Nⁿ. Then, for h>0small enough, there exists a bijection bh between the sets Res0(P)∩D(E0,Ch)andRes(P)∩D(E0,Ch)counted with their multiplicity, where

Res0(P) =n

z_α⁰ =E0−ih

n

X

j=1

αj+1 2

λj; α∈Nⁿ o

,

such that bh(z)−z =o(h).

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Quantum resonances:

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A homoclinic orbit in 1d

I We suppose here thatd= 1 and that

x E0

0 V(x)

H (0,0) p⁻¹(E₀)⊂T^∗Rⁿ

Theorem (Fujii´ e–R.)

The resonances(zk)_k∈Nof P in D(E0,Ch)satisfies z_k =E₀+λS₀−(2k+ 1)πh

2|lnh| −i h

|lnh|

λln 2

2 +O h/(lnh)² , where V(x) =E₀−^λ₂²x²+O(x³).

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Quantum resonances:

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I The Time delay:

V0 E

I ForE ∈]0,+∞[,S(E,h) is unitary, and the scattering phase is defined as θ(E,h) = _2i¹ln detS(E,h).

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Quantum resonances:

a brief survey

Introduction Mathematical definitions Resonances in 1d Some examples The trapped set

Settings Definition and examples When the trapped set is empty Semiclassical resolvent estimates An application : the quantum evolution

Settings

I We consider the semiclassical Schr¨odinger operator onRⁿ, n≥1,

P=−h²∆ +V(x) where V(x)∈C^∞(Rⁿ;R).

I We define the resonances by analytic distortion, assuming that V(x) has an analytic extension to

Σ ={x∈Cⁿ; |Imx|< δ|Rex|and|Rex|>C}

for some δ,C>0, andV(x)−→0 as|x| → ∞,x∈Σ.

Rⁿ Cⁿ

C arctanδ

Σ

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Quantum resonances:

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The trapped set

I We have seen three situations where one has been able to show existence and give precise location, in the semiclassical regime, of resonances close to someE₀∈R. In these three cases, the trapped set at energyE₀is not empty :

Definition :

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

(x, ξ)∈p⁻¹(E₀); t7→exp(tH_p)(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 vector field onT^∗Rⁿ=R²ⁿ given byH_p=∂_ξp∂_x−∂_xp∂_ξ=

2ξ

−∇V(x)

.

I Examples :

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

- A barrier top - The homoclinic case

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Quantum resonances:

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What if the trapped set is empty ?

Theorem (Martinez)

Suppose K(E0) =∅. Then P has no resonance in Zh(E0) = [E0−ε,E0+ε] +i[−C h|lnh|,0]

for some ε >0and all C >0. Moreover, for allχ∈C₀^∞(Rⁿ), there exists N =N(C)such that for z∈Zh(E0),

kχ(P−z)⁻¹χk.h^−N.

I IfV is analytic in a whole neighborhood ofRⁿ,

Briet–Combes–Duclos and Helffer–Sj¨ostrand have proved thatP has no resonance in a fixed neighborhood ofE0.

I Notice that ifχ∈ C₀^∞(Rⁿ) is supported where the distortion opertor does nothing, thenχ(P−z)⁻¹χ=χ(Pθ−z)⁻¹χ, so that the estimate on the resolvent implies that there is no resonance in Z_h(E₀).

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Quantum resonances:

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Resolvent estimate for real energies

Theorem

For E0>0, the following properties are equivalent.

i) K(E0) =∅.

ii) There existsε >0 such that, for anyχ∈C₀^∞(Rⁿ), sup

z∈[E0−ε,E0+ε]

χ(P−z)⁻¹χ .h⁻¹.

Theorem (Bony-Burq-R)

i) K(E₀)6=∅.

ii) There existsχ∈C₀^∞(Rⁿ) such that, for allε >0, sup

z∈[E0−ε,E0+ε]

χ(P−z)⁻¹χ

&h⁻¹|lnh|.

Moreover, when K(E₀)6=∅, the above estimate is true for any χ∈C₀^∞(Rⁿ) such thatχ= 1 nearπ_xK(E₀).

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Quantum resonances:

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Resolvent estimate for real energies

Theorem

i) K(E0) =∅.

ii) There existsε >0 such that, for anyχ∈C₀^∞(Rⁿ), sup

z∈[E0−ε,E0+ε]

χ(P−z)⁻¹χ .h⁻¹.

Theorem (Bony-Burq-R)

i) K(E₀)6=∅.

ii) There existsχ∈C₀^∞(Rⁿ) such that, for allε >0, sup

z∈[E0−ε,E0+ε]

χ(P−z)⁻¹χ

&h⁻¹|lnh|.

Moreover, when K(E₀)6=∅, the above estimate is true for any χ∈C₀^∞(Rⁿ) such thatχ= 1 nearπ_xK(E₀).

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Quantum resonances:

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Proof (1)

I (ii)⇒(i) by the first theorem. We only have to prove that (i)⇒

(ii). We set

h⁻¹M(h) = sup

z∈[E0−ε,E0+ε]

χ(P−z)⁻¹χ .

Using Kato smoothness, we have Z

R

χ1_[E₀_−ε,E₀_+ε](P)e^isPu

2

L²ds.h⁻¹M(h)kuk². Therefore, ifϕ∈C₀^∞([E0−ε,E0+ε]; [0,1]) with ϕ(E0) = 1,

Z

R

χϕ(P)e^itP/hu

2

L²dt.M(h)kuk².

I Nowϕ(P)χ²ϕ(P) =Op^w_h(a), where

a(x, ξ;h) =a₀(x, ξ) +hS_h(1), a₀(x, ξ) =χ²(x)ϕ(p(x, ξ))². Thus we have

Z

R

Op^w_h(a)e^itP/hu,e^itP/hu

dt.M(h)kuk².

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Quantum resonances:

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Proof (2)

I Now letu=uρ be a normalized gaussian coherent state centered atρ∈K(E₀) (u=T_ρ^h(e^−x²^/2h)). We have seen in Clotilde’s talk that

h→0lim Op^w_h(a)e^itP/hu_ρ,e^itP/hu_ρ

=a₀(exp(tH_p)(ρ)), for|t| ≤C|lnh|, whereC >0.

I Since exp(tHp)∈K(E0) for allt, we havea0(exp(tHp)(ρ)) = 1, and

Op^w_h(a)e^itP/hu_ρ,e^itP/hu_ρ

≥1 2, provided his small enough. We obtain

C|lnh| ≤ Z

|t|≤C|lnh|

Op^w_h(a)e^itP/hu,e^itP/hu

dt.M(h)kuk². so that, as stated,

|lnh|.M(h).

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Quantum resonances:

a brief survey

Introduction Mathematical definitions Resonances in 1d Some examples The trapped set Semiclassical resolvent estimates

A universal upper bound The barrier top case An application : the quantum evolution

A universal upper bound

Theorem (Burq)

For a broad class of operators P, whatever the trapped set is, there is no resonance in

[E₀−ε,E₀+ε] +ih

e^−C^/h,0i for some ε,C >0. Moreover

sup

z∈[E0−ε,E0+ε]

χ(P−z)⁻¹χ

.e^C/h.

I This theorem is optimal, as shown by the ”Puits dans l’isle”

case.

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Quantum resonances:

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The barrier top case

I We suppose thatV has an analytic extension to Σ ={x∈Cⁿ; |Imx|< δhRexi}

for some δ >0, and thatV(x)−→0 as |x| → ∞, x∈Σ.

Moreover, we assume

V(x) =E0−

n

X

j=1

λ²_jx_j²+O(|x|³).

and that K(E0) ={0}.

I We already know that there existsδ >0 such thatPhas no resonance inD(E0, δh).

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Quantum resonances:

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Resolvent estimates in the barrier top case

Theorem (Alexandrova-Bony-R)

There existsε >0 such that, for anyχ∈ C₀^∞(Rⁿ), and for

|z −E0| ≤ε.

kχ(P−z)⁻¹χk.h⁻¹|lnh|,

Theorem (Bony-Fujii´ e-R-Zerzeri)

There existsε >0 such that, for allC >0 andhsmall enough, i) The operator Phas no resonances in

[E0−ε,E0+ε] +i[−Ch,0]\D(E0,2Ch).

ii) There existsK >0 such that χ(P−z)⁻¹χ

.h^−K Y

z_α∈Res(P)∩D(E0,2Ch)

|z−zα|⁻¹,

for allz ∈[E0−ε,E0+ε] +i[−Ch,Ch].

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Quantum resonances:

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Resolvent estimates in the barrier top case

Theorem (Alexandrova-Bony-R)

There existsε >0 such that, for anyχ∈ C₀^∞(Rⁿ), and for

|z −E0| ≤ε.

kχ(P−z)⁻¹χk.h⁻¹|lnh|,

Theorem (Bony-Fujii´ e-R-Zerzeri)

There existsε >0 such that, for allC >0 andhsmall enough, i) The operator Phas no resonances in

[E0−ε,E0+ε] +i[−Ch,0]\D(E0,2Ch).

ii) There existsK >0 such that χ(P−z)⁻¹χ

.h^−K Y

z_α∈Res(P)∩D(E0,2Ch)

|z−zα|⁻¹,

for allz ∈[E0−ε,E0+ε] +i[−Ch,Ch].

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Quantum resonances:

a brief survey

The shape resonances case

The barrier top case

The Schr¨ odinger group

I We denote bye^−itP/hu₀the solutionu to the time-dependent Schr¨odinger equation

i~∂_tφ(t,x) =P(x,~D_x)φ(t,x), u_|_t=0 =u₀.

I Ifλ₀∈Ris an isolated eigenvalue of P, and

ψ∈ C₀^∞(]λ₀−δ, λ₀+δ[) for someδ >0 small enough, one has e^−itP/hψ(P) =e^−itλ⁰^/hΠ_λ₀ψ(λ₀),

where Πλ₀ is the spectral (orthogonal) projection onto the eigenspace associated to λ0.

I Is (how) the behavior of the quantum evolution related to the resonances ?

(37)

Quantum resonances:

a brief survey

The spectral projection for a resonance

Definition

Letz be a resonance for the opertorP. The associated (generalized) spectral projection is

Πz=− 1 2iπ

I

γz

(P−ζ)⁻¹dζ,

as an operator fromL²_comp toL²_loc. The multiplicity of a resonance is the rank of Π_z.

I Notice that Πz is not an orthogonal projection.

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Quantum resonances:

a brief survey

The shape resonances case

I When the resonances are close to the real axis, one can expect that they truly influence the quantum evolution. In the case of shape resonances we have the

Theorem : (Nakamura-Stefanov-Zworski)

Letχ∈ C₀^∞(Rⁿ) and writeχ=χ₁+χ₂withχ₂not supported near the well, and χ₁not supported near the sea. Let

ψ∈C₀^∞([E₀−ε,E₀+ε]) for someε >0 small enough. Then for anyµ >0, there existsT >0 such that

χe^−itP/hχψ(P) = X

z_α∈Res(P)∩D(E₀,µh)

e^−itz^α^/hχ₁Π_z_αχ₁ψ(P)

+O(h^∞) +χ₂O(((t−T)₊

h )^−∞)χ₂, for allt ≥0.

I Notice that this makes sense only fort >T.

(39)

Quantum resonances:

a brief survey

The barrier top case

I Suppose that we are in the barrier-top situation. The resonances zαare close to thez_α⁰ =E0−ihPn

j=1(αj+¹₂)λj.

Theorem : (Bony-Fujiie-R-Zerzeri)

Letµ >0 be different fromPn

j=1(αj+¹₂)λj for allα∈Nⁿ. Suppose that all the z_α⁰ inD(E0, µh) are simple.

Letχ∈C₀^∞(Rⁿ) and ψ∈C₀^∞([E0−ε,E0+ε]) for someε >0 small enough.

Then, there existsK =K(µ)>0 such that

χe^−itP/hχψ(P) = X

z_α∈Res(P)∩D(E0,µh)

e^−itz^α^/hχΠzαχψ(P) +O(h^∞) +O(e^−µth^−K), for allt ≥0.

I This makes sense only fort >K|lnh|. Whent/|lnh| →+∞as h→0, the sum over the resonances is negligible and this result says onlyχe^−itP/hχψ(P) =O(h^∞).