On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime

Weizhu Bao*, Shi Jin, Peter Markowich

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

277 Scopus citations

Abstract

In this paper we study time-splitting spectral approximations for the linear Schrödinger equation in the semiclassical regime, where the Planck constant e is small. In this regime, the equation propagates oscillations with a wavelength of O (e), and finite difference approximations require the spatial mesh size h = o (e) and the time step k = o (e) in order to obtain physically correct observables. Much sharper mesh-size constraints are necessary for a uniform L2-approximation of the wave function. The spectral time-splitting approximation under study will be proved to be unconditionally stable, time reversible, and gauge invariant. It conserves the position density and gives uniform L2-approximation of the wave function for k = o (e) and h = O (e). Extensive numerical examples in both one and two space dimensions and analytical considerations based on the Wigner transform even show that weaker constraints (e.g., k independent of e, and h = O (e)) are admissible for obtaining "correct" observables. Finally, we address the application to nonlinear Schrödinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies.

Original languageEnglish (US)
Pages (from-to)487-524
Number of pages38
JournalJournal of Computational Physics
Volume175
Issue number2
DOIs
StatePublished - Jan 20 2002

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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