Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime

Abstract

We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrödinger equation. Traditional Gaussian beam type methods for the Schrödinger 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 a new 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 Schrödinger 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 Schrödinger equation. For instance, in the one-dimensional case the proposed algorithm requires only O ( sNm 2 ) operations to compute s different solutions with s different initial wave functions under the influence of the same potential, where N = O ( 1 / ℏ ) , ℏ is the Planck constant, and m ≪ N is the number of computed beams which depends on ℏ weakly. Numerical experiments indicate that this Eulerian Gaussian beam approach yields accurate semi-classical solutions even at caustics.