Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary

Abstract

This article describes how high-frequency waves solutions to the scalar wave equation and the Schrödinger equation propagate — in terms of semiclassical measures (also called Wigner measures) — across a sharp interface between two inhomogeneous media. A microlocal version of Snell–Descartes's law of refraction which includes diffractive rays is proved. A radiation phenomenon for waves density propagating inside an interface along gliding rays is illustrated. The underlying results on semiclassical measures are presented in the general context of a local second order semiclassical partial differential equation problem in which some a priori bounds on the solutions and its traces allow to consider their semiclassical measures. Thanks to the measures of the traces, some boundary conditions and propagation properties for the measure of the solutions are derived. These results also yield propagation laws — such as damped reflection and damped propagation in the boundary — for other boundary value problems.