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Mathematical and computational methods for semiclassical Schrödinger equations*

Published online by Cambridge University Press:  28 April 2011

Shi Jin ,
Peter Markowich  and
Christof Sparber
Shi Jin
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA E-mail: jin@math.wisc.edu
Peter Markowich
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK E-mail: p.markowich@damtp.cam.ac.uk
Christof Sparber
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607, USA E-mail: sparber@math.uic.edu
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We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical regime.

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Type
Research Article
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Acta Numerica , Volume 20 , May 2011 , pp. 121 - 209
Copyright
Copyright © Cambridge University Press 2011

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