## 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 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 (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 language | English (US) |
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Pages (from-to) | 487-524 |

Number of pages | 38 |

Journal | Journal of Computational Physics |

Volume | 175 |

Issue number | 2 |

DOIs | |

State | Published - 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