Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation

  • Authors:

  • Jianliang Qian;Lexing Ying

  • Affiliations:

  • Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States;Department of Mathematics and ICES, University of Texas at Austin, Austin, TX 78712, United States

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2010

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Abstract

This paper introduces a wavepacket-transform-based Gaussian beam method for solving the Schrodinger equation. We focus on addressing two computational issues of the Gaussian beam method: how to generate a Gaussian beam representation for general initial conditions and how to perform long time propagation for any finite period of time. To address the first question, we introduce fast Gaussian wavepacket transforms and develop on top of them an efficient initialization algorithm for general initial conditions. Based on this new initialization algorithm, we address the second question by reinitializing the beam representation when the beams become too wide. Numerical examples in one, two, and three dimensions demonstrate the efficiency and accuracy of the proposed algorithms. The methodology can be readily generalized to deal with other semi-classical quantum mechanical problems.