Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow

Combescure, Monique; Robert, Didier

doi:10.3233/asy-1997-14405

Cited by 91 publications

(135 citation statements)

References 25 publications

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“…Now we compute U (t)ϕ α , using the semiclassical propagation of coherent states result as it was formulated in Combescure-Robert [10]. We recall that F (t) is a time dependent symplectic matrix (Jacobi matrix) defined by the linear equation (9).…”

Section: Preparations For the Proofmentioning

confidence: 99%

A Proof of the Gutzwiller Semiclassical Trace Formula Using Coherent States Decomposition

Combescure¹,

Ralston²,

Robert³

1999

Communications in Mathematical Physics

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The Gutzwiller trace formula links the eigenvalues of the Schrödinger operator H as Planck's constant goes to zero (the semiclassical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral representation for the propagator of H. Later, using the theory of Fourier integral operators, mathematicians gave rigorous proofs of the formula in various settings. Here we show how the use of coherent states allows us to give a simple and direct proof.

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Section: Preparations For the Proofmentioning

confidence: 99%

A Proof of the Gutzwiller Semiclassical Trace Formula Using Coherent States Decomposition

Combescure¹,

Ralston²,

Robert³

1999

Communications in Mathematical Physics

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“…Now v 2 will be purely repulsive and for this reason the h 2 flow will not have any unstable fixed points. In this way the unstable fixed point long time breakdown [28,34,17] of the standard semiclassical approximation is avoided.…”

Section: Discussionmentioning

confidence: 99%

Mixed Weyl symbol calculus and spectral line shape theory

Osborn

Kondrat'eva

Tabisz

et al. 1999

J. Phys. A: Math. Gen.

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A new and computationally viable full quantum version of line shape theory is obtained in terms of a mixed Weyl symbol calculus. The basic ingredient in the collision-broadened line shape theory is the time dependent dipole autocorrelation function of the radiator-perturber system. The observed spectral intensity is the Fourier transform of this correlation function. A modified form of the Wigner-Weyl isomorphism between quantum operators and phase space functions (Weyl symbols) is introduced in order to describe the quantum structure of this system. This modification uses a partial Wigner transform in which the radiator-perturber relative motion degrees of freedom are transformed into a phase space dependence, while operators associated with the internal molecular degrees of freedom are kept in their original Hilbert space form. The result of this partial Wigner transform is called a mixed Weyl symbol. The star product, Moyal bracket and asymptotic expansions native to the mixed Weyl symbol calculus are determined. The correlation function is represented as the phase space integral of the product of two mixed symbols: one corresponding to the initial configuration of the system, the other being its time evolving dynamical value. There are, in this approach, two semiclassical expansionsone associated with the perturber scattering process, the other with the mixed symbol star product. These approximations are used in combination to obtain representations of the autocorrelation that are sufficiently simple to allow numerical calculation. The leading O(h 0 ) approximation recovers the standard classical path approximation for line shapes. The higher order O(h 1 ) corrections arise from the noncommutative nature of the star product.

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“…Denote by Ψ the orthogonal projector on ϕ 0 , Ψu := u, ϕ 0 ϕ 0 ; it is a pseudodifferential operator with -Weyl symbol given by the Wigner function W ϕ0 (x, ξ) = 2 d e − |x| 2 +|ξ| 2 , see e.g. [11]. Now remark that for any operator A one has Aϕ 0 , ϕ 0 = AΨϕ 0 , ϕ 0 = j≥0 AΨϕ j , ϕ j = tr(AΨ),…”

Section: )mentioning

confidence: 99%

“…The second step is to show that there exists a solution of (1.1) which stays close, in the H r topology, to such coherent state for all times in (1.7). This is done by extending classical results of semiclassical approximation [20,11] to the H r topology, a result which we think might be interesting in its own. Theorem 1.1 extends partially to anharmonic oscillators a result of [3], which, in case of the quantum harmonic oscillators on R d , constructs solutions with unbounded path in Sobolev spaces.…”

Section: Introduction and Statementmentioning

confidence: 99%