Past Event: Oden Institute Seminar

Eulerian Gaussian beams for semi-classical solutions of Schrodinger equations

Jianliang Qian, Professor, Michigan State University

Abstract

We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrodinger equation. Traditional Gaussian beam type methods for the Schrodinger equation are based on the Lagrangian ray tracing. Based on the first Eulerian Gaussian beam framework proposed in Leung et al. [S. Leung, J. Qian, R. Burridge, Eulerian Gaussian beams for high frequency wave propagation, Geophysics 72 (2007) SM61-SM76], we develop an Eulerian Gaussian beam method which uses global Cartesian coordinates, level-set based implicit representation and Liouville equations. The resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously. To obtain semi-classical solutions to the Schrodinger equation with different initial wave functions, we only need to slightly modify the summation formula. This yields a very efficient method for computing semi-classical solutions to the Schrodinger equation. We also highlight the importance of initializing Gaussian beam propagation. Numerical experiments indicate that this Eulerian Gaussian beam approach yields accurate semi-classical solutions even at caustics. This is a joint work with S. Leung and R. Burridge.