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PROPAGATION OF SEMICLASSICAL WAVE PACKETS THROUGH AVOIDED EIGENVALUE CROSSINGS IN NONLINEAR SCHRÖDINGER EQUATIONS

Part of: Equations of mathematical physics and other areas of application Groups and algebras in quantum theory General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  20 October 2014

Lysianne Hari
Lysianne Hari*
Affiliation:
University of Cergy-Pontoise, UMR CNRS 8088, F-95000 Cergy-Pontoise, France ([email protected])
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Abstract

We study the propagation of wave packets for a one-dimensional system of two coupled Schrödinger equations with a cubic nonlinearity, in the semiclassical limit. Couplings are induced by the nonlinearity and by the potential, whose eigenvalues present an avoided crossing: at one given point, the gap between them reduces as the semiclassical parameter becomes smaller. For data which are coherent states polarized along an eigenvector of the potential, we prove that when the wave function propagates through the avoided crossing point there are transitions between the eigenspaces at leading order. We analyze the nonlinear effects, which are noticeable away from the crossing point, but see that in a small time interval around this point the nonlinearity’s role is negligible at leading order, and the transition probabilities can be computed with the linear Landau–Zener formula.

Keywords

nonlinear Schrödinger equationssemiclassical limitquantum propagation of coherent statesavoided crossing

MSC classification

Primary: 35Q40: PDEs in connection with quantum mechanics
Secondary: 35Q55: NLS-like equations (nonlinear Schrödinger) 81Q05: Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 81R20: Covariant wave equations 81R30: Coherent states
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 2 , April 2016 , pp. 319 - 365
Copyright
© Cambridge University Press 2014 

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