Mathematical and Numerical Methods for Complex Quantum Systems

Frozen Gaussian approximation for wave propagation in periodic media

Xu Yang

University of California, Santa Barbara

Unlike the Gaussian beam method (GBM), frozen Gaussian approximation (FGA) is a semiclassical approximation to high frequency waves with Gaussians of fixed width. It was called Herman-Kluk approximation in quantum chemistry, and used for solving the Schrodinger equation. In this talk, we generalize the idea, derive and analyze the accuracy of FGA for waves propagating in periodic media, modeled by the Schrodinger equation with a lattice potential. Assuming the existence of band gap, we are able to obtain the effective dynamic equations of electrons within each energy band, and prove the first order accuracy of FGA.