Mathematical and computational methods for semiclassical Schrödinger equations
journal contributionposted on 2013-12-12, 00:00 authored by Shi Jin, Peter Markowich, Christof Sparber
We consider time-dependent (linear and nonlinear) Schr¨odinger 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 Schr¨odinger equation in the semiclassical regime.
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 University of Wisconsin–Madison. Supported by a Royal Society Wolfson Research Merit Award and by KAUST through a Investigator Award KUK-I1-007-43. Partially supported by the Royal Society through a University Research Fellowship.
Publisher StatementThis is a copy of an article published in the Acta Numerica © 2011 Cambridge University Press. The final publication is available at http://journals.cambridge.org/abstract_S0962492911000031
PublisherCambridge University Press