Mathematical and computational methods for semiclassical Schrödinger equations

Shi Jin, Peter A. Markowich, Christof Sparber

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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. © 2011 Cambridge University Press.
Original languageEnglish (US)
Pages (from-to)121-209
Number of pages89
JournalActa Numerica
StatePublished - Apr 28 2011
Externally publishedYes

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