Bohr-Sommerfeld Rules to All Orders

Verdìère, Yves Colin de

doi:10.1007/s00023-005-0230-z

Cited by 53 publications

(44 citation statements)

References 10 publications

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“…This Theorem is closely related to (but a bit stronger) than what is proved in my paper [3]. The trace formula contains implicitely the Maslov index.…”

supporting

confidence: 63%

“…This can be seen as a complement and a partial rewriting of my paper [3] with a better trace formula. The formula we will prove is more general than that in Section 6.…”

Section: Appendix A: Abel's Resultsmentioning

confidence: 99%

“…The case where F is compactly supported: using Poisson summation formula as in [3], we get TraceF (Ĥ) = 1 2π F (y)S ′ (y)dy and we get this case by integration by part.…”

mentioning

confidence: 98%

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A semi-classical inverse problem II: reconstruction of the potential

Verdìère

2011

Progress in Mathematics

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“…This Theorem is closely related to (but a bit stronger) than what is proved in my paper [3]. The trace formula contains implicitely the Maslov index.…”

supporting

confidence: 63%

“…This can be seen as a complement and a partial rewriting of my paper [3] with a better trace formula. The formula we will prove is more general than that in Section 6.…”

Section: Appendix A: Abel's Resultsmentioning

confidence: 99%

See 1 more Smart Citation

A semi-classical inverse problem II: reconstruction of the potential

Verdìère

2011

Progress in Mathematics

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“…Remark 3.2 In recent papers [6], [21] the authors have given some methods to compute explicitly the terms b j for j ≥ 2 in the expansion in in the Bohr-Sommerfeld rule.…”

Section: Revivals For 1-d Systemsmentioning

confidence: 99%

A Phase-Space Study of the Quantum Loschmidt Echo in the Semiclassical Limit

Combescure

Robert

2006

Ann. Henri Poincaré

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The notion of Loschmidt echo (also called "quantum fidelity") has been introduced in order to study the (in)-stability of the quantum dynamics under perturbations of the Hamiltonian. It has been extensively studied in the past few years in the physics literature, in connection with the problems of "quantum chaos", quantum computation and decoherence. In this paper, we study this quantity semiclassically (as → 0), taking as reference quantum states the usual coherent states. The latter are known to be well adapted to a semiclassical analysis, in particular with respect to semiclassical estimates of their time evolution. For times not larger than the so-called "Ehrenfest time" C| log |, we are able to estimate semiclassically the Loschmidt Echo as a function of t (time), (Planck constant), and δ (the size of the perturbation). The way two classical trajectories merging from the same point in classical phase-space, fly apart or come close together along the evolutions governed by the perturbed and unperturbed Hamiltonians play a major role in this estimate. We also give estimates of the "return probability" (again on reference states being the coherent states) by the same method, as a function of t and .

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“…The strategy to understand the long times behaviour of dynamics is to use the spectrum of the operator P h . In the regular case, the spectrum of P h is given by the famous Bohr-Sommerfeld rules (see, for example, [5,7,13]): in first approximation, the spectrum of P h in a compact set is a sequence of real numbers with a gap of size h. The classical trajectories are periodic and supported on elliptic curves.…”

Section: Context and Motivationmentioning

confidence: 99%

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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Bohr-Sommerfeld Rules to All Orders

Cited by 53 publications

References 10 publications

A semi-classical inverse problem II: reconstruction of the potential

A semi-classical inverse problem II: reconstruction of the potential

A Phase-Space Study of the Quantum Loschmidt Echo in the Semiclassical Limit

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

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