Mathematical and computational methods for semiclassical Schrödinger equations

Shi Jin*, Peter Markowich, Christof Sparber

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

113 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)121-209
Number of pages89
JournalActa Numerica
Volume20
DOIs
StatePublished - Apr 2011

ASJC Scopus subject areas

  • Numerical Analysis
  • General Mathematics

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