Sur le spectre semi-classique d’un système intégrable de dimension 1 autour d’une singularité hyperbolique

Lablée, Olivier

doi:10.5802/tsg.260

Cited by 4 publications

(10 citation statements)

References 18 publications

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“…Thus, we do not observe the expected increase in the size of the gaps which means that this transition does not hold in intervals like [E 0 − Ch, E 0 + Ch]. This is not surprising since the eigenvalues for homoclinic orbits in dimension n = 1 satisfy similar properties (see Lablée [66]).…”

Section: Remarkable Phenomenamentioning

confidence: 69%

Resonances for homoclinic trapped sets

Bony,

Fujiie,

Ramond

et al. 2016

Preprint

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We study semiclassical resonances generated by homoclinic trapped sets. First, under some general assumptions, we prove that there is no resonance in a region below the real axis. Then, we obtain a quantization rule and the asymptotic expansion of the resonances when there is a finite number of homoclinic trajectories. The same kind of results is proved for homoclinic sets of maximal dimension. Next, we generalize to the case of homoclinic/heteroclinic trajectories and we study the three bump case. In all these settings, the resonances may either accumulate on curves or form clouds. We also describe the corresponding resonant states.

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Section: Remarkable Phenomenamentioning

confidence: 69%

Resonances for homoclinic trapped sets

Bony,

Fujiie,

Ramond

et al. 2016

Preprint

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“…Record here this description. We take the presentation from the paper [16] and refer to this paper for more details. [16])…”

Section: Spectrum Near the Singularitymentioning

confidence: 99%

“…In [8][9][10]16] we have the smooth function E → ε(E) defined for |E| δ, where δ is a real constant and not depends to h; (this function is linked to a normal form [6,8]) defined by…”

Section: Some Notations (Seementioning

confidence: 99%

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Semi-classical behaviour of Schrödinger's dynamics: Revivals of wave packets on hyperbolic trajectory

Lablée¹

2011

Asymptotic Analysis

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The aim of this paper is to study the semi-classical behaviour of Schrödinger's dynamics for an one-dimensional quantum Hamiltonian with a classical hyperbolic trajectory. As in the regular case (elliptic trajectory), we prove, that for an initial wave packets localized in energy, the dynamics follows the classical motion during short time. This classical motion is periodic and the period T hyp is order of | ln h|. And, for large time, a new period Trev for the quantum dynamics appears: the initial wave packets form again at t = Trev. Moreover, for the time t = p q Trev a fractional revivals phenomenon of the initial wave packets appears: there is a formation of a finite number of clones of the original wave packet. Keywords: Schrödinger's dynamics, revivals of wave packets, semi-classical analysis, hyperbolic trajectory, Schrödinger operator with double wells potential 0921-7134/11/$27.50 © 2011 -IOS Press and the authors. All rights reserved O. Lablée / Semi-classical behaviour of Schrödinger's dynamics 61 2. Quantum dynamics and autocorrelation function 2.1. The quantum dynamics For a quantum Hamiltonian P h : D(P h ) ⊂ H → H, H is a Hilbert space, the Schrödinger dynamics is governed by the Schrödinger equation ih ∂ψ(t) ∂t = P h ψ(t). With the functional calculus, we can reformulate this equation with the unitary group UIndeed, for a initial state ψ 0 ∈ H, the evolution given by Return and autocorrelation functionWe now introduce a simple tool to understand the behaviour of the vector ψ(t): a quantum analog of the Poincaré return function.Definition. The quantum return function of the operator P h and for an initial state ψ 0 is defined by r(t) := ψ(t), ψ 0 H , and the autocorrelation function is defined byThe previous function measures the return on initial state. This function is the overlap of the time dependent quantum state ψ(t) with the initial state ψ 0 . Since the initial state ψ 0 is normalized, the autocorrelation function takes values in the compact set [0, 1]. Then, if we have an orthonormal basis of eigenvectors (e n ) n∈N :we get, for all integer nSo for a initial vector ψ 0 ∈ D(P h ) ⊂ H, let us denote by (c n ) n∈N = (c n (h)) n∈N the sequence of 2 (N) given (c n ) n = π(ψ 0 ), where π is the projector (unitary operator):π : H → 2 (N), ψ → ψ, e n H .

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“…The spectrum near this singularity is more complicated than in the regular case. In [Lab3] we have an explicit description of the spectrum for an one-dimesional pesudo-differential operator near a hyperbolic non-degenerate singularity. The article [Lab4] deals with the quantum dynamics for the hyperbolic case.…”

Section: Introductionmentioning

confidence: 99%

Quantum revivals in two degrees of freedom integrable systems: The torus case

Lablée

2012

Asymptotic Analysis

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The paper deals with the semi-classical behaviour of quantum dynamics for a semi-classical completely integrable system with two degrees of freedom near Liouville regular torus. The phenomomenon of wave packet revivals is demonstrated in this article. The framework of this paper is semi-classical analysis (limit : h → 0). For the proofs we use standard tools of real analysis, Fourier analysis and basic analytic number theory.

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Sur le spectre semi-classique d’un système intégrable de dimension 1 autour d’une singularité hyperbolique

Cited by 4 publications

References 18 publications

Resonances for homoclinic trapped sets

Resonances for homoclinic trapped sets

Semi-classical behaviour of Schrödinger's dynamics: Revivals of wave packets on hyperbolic trajectory

Quantum revivals in two degrees of freedom integrable systems: The torus case

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