On Time-Splitting Spectral Approximations for the Schrödinger Equation in the Semiclassical Regime

✍ Scribed by Weizhu Bao; Shi Jin; Peter A. Markowich

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 942 KB
• Volume: 175
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.2001.6956

✦ Synopsis

In this paper we study time-splitting spectral approximations for the linear Schrödinger equation in the semiclassical regime, where the Planck constant ε is small. In this regime, the equation propagates oscillations with a wavelength of O(ε), and finite difference approximations require the spatial mesh size h = o(ε) and the time step k = o(ε) in order to obtain physically correct observables. Much sharper mesh-size constraints are necessary for a uniform L 2 -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 L 2 -approximation of the wave function for k = o(ε) and h = O(ε). 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 ε, and h = O(ε)) 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.