Title: Wave dynamics in locally periodic structures by multiscale analysis
Description: We study the propagation of waves in spatially non-homogeneous media focusing on Schrodinger’s equation of quantum mechanics and Maxwell’s equations of electromagnetism. We assume that medium variation occurs over two distinct length scales: a short ‘fast’ scale with respect to which the variation is periodic, and a long ‘slow’ scale over which the variation is smooth. Let epsilon denote the ratio of these scales. We focus primarily on the time evolution of asymptotic solutions (as epsilon tends to zero) known as semiclassical wavepackets. Such solutions generalize exact time-dependent Gaussian solutions and ideas of Heller and Hagedorn to periodic media. Our results are as follows: 1) To leading order in epsilon and up to the ‘Ehrenfest’ time-scale t ~ log 1/epsilon, the center of mass and average (quasi-)momentum of the semiclassical wavepacket satisfy the equations of motion of the classical Hamiltonian given by the wavepacket’s Bloch band energy. Our first result is to derive all corrections to these dynamics proportional to epsilon.

These corrections consist of terms proportional to the Bloch band’s Berry curvature and terms which describe coupling to the evolution of the wavepacket envelope. These results rely on the assumption that the wavepacket’s Bloch band energy is non-degenerate. 2) We then consider the case where, in one spatial dimension, a semiclassical wavepacket is incident on a Bloch band crossing, a point in phase space where the wavepacket’s Bloch band energy is degenerate. By a rigorous matched asymptotic analysis, we show that at the time the wavepacket meets the crossing point a second wavepacket, associated with the other Bloch band involved in the crossing, is excited. Our result can be seen as a rigorous justification of the Landau-Zener formula in this setting. 3) Our final result generalizes the recent work of Fefferman, Lee-Thorp, and Weinstein on one-dimensional ‘edge’ states. We characterize the bound states of a Schrodinger operator with a periodic potential perturbed by multiple well-separated domain wall ‘edge’ modulations, by proving a theorem on the near zero eigenstates of an emergent Dirac operator.