Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials

Pages: 759 - 774, Volume 17, Issue 3, May 2012

doi:10.3934/dcdsb.2012.17.759       Abstract        References        Full Text (463.4K)       Related Articles

Rémi Carles - CNRS & Univ. Montpellier 2, Mathématiques, CC 051, 34095 Montpellier, France (email)
Christof Sparber - Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, 851 South Morgan Street, Chicago, Illinois 60607, United States (email)

Abstract: We consider semiclassically scaled Schrödinger equations with an external potential and a highly oscillatory periodic potential. We construct asymptotic solutions in the form of semiclassical wave packets. These solutions are concentrated (both, in space and in frequency) around the effective semiclassical phase-space flow, and involve a slowly varying envelope whose dynamics is governed by a homogenized Schrödinger equation with time-dependent effective mass tensor. The corresponding adiabatic decoupling of the slow and fast degrees of freedom is shown to be valid up to Ehrenfest time scales.

Keywords:  Schrödinger equation, semiclassical wave packet, periodic potential, Bloch band, Ehrenfest time.
Mathematics Subject Classification:  81Q20, 35A35, 35Q40, 81Q05.

Received: March 2011;      Revised: November 2011;      Published: January 2012.

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