Mathematical and computational methods for semiclassical Schrödinger equations

Type
Article

Authors
Jin, Shi
Markowich, Peter A.
Sparber, Christof

KAUST Grant Number
KUK-I1-007-43

Online Publication Date
2011-04-28

Print Publication Date
2011-05

Date
2011-04-28

Abstract
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. © 2011 Cambridge University Press.

Citation
Jin S, Markowich P, Sparber C (2011) Mathematical and computational methods for semiclassical Schrödinger equations. Acta Numerica 20: 121–209. Available: http://dx.doi.org/10.1017/s0962492911000031.

Acknowledgements
Partially supported by NSF grant no. DMS-0608720, NSF FRG grant DMS-0757285,a Van Vleck Distinguished Research Prize and a Vilas Associate Award from the Universityof Wisconsin–Madison.Supported by a Royal Society Wolfson Research Merit Award and by KAUST througha Investigator Award KUK-I1-007-43.Partially supported by the Royal Society through a University Research Fellowship.

Publisher
Cambridge University Press (CUP)

Journal
Acta Numerica

DOI
10.1017/s0962492911000031

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