Gauge-invariant frozen Gaussian approximation method for the Schr\"odinger equation with periodic potentials

We develop a gauge-invariant frozen Gaussian approximation (GIFGA) method for the linear Schr\"odinger equation (LSE) with periodic potentials in the semiclassical regime. The method generalizes the Herman-Kluk propagator for LSE to the case with periodic media. It provides an efficient computational tool based on asymptotic analysis on phase space and Bloch waves to capture the high-frequency oscillations of the solution. Compared to geometric optics and Gaussian beam methods, GIFGA works in both scenarios of caustics and beam spreading. Moreover, it is invariant with respect to the gauge choice of the Bloch eigenfunctions, and thus avoids the numerical difficulty of computing gauge-dependent Berry phase. We numerically test the method by several one-dimensional examples, in particular, the first order convergence is validated, which agrees with our companion analysis paper [Delgadillo, Lu and Yang, arXiv:1504.08051].