Title: Fast Gaussian Wavepacket Transforms and Gaussian Beams for the
Schr\"odinger Equation

Speaker: Jianliang Qian, Department of Mathematics, Michigan State
University, East Lansing, MI 48824.

Abstract:

We introduce a wavepacket-transform-based Gaussian beam method for
solving the Schr\"odinger 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.