In-out decomposition of boundary integral equations

TL;DR

This paper introduces an exact boundary integral reformulation for the Helmholtz equation using incoming and outgoing waves, enabling precise analysis of wave phenomena like diffraction and evanescent coupling beyond semiclassical approximations.

Contribution

It presents a boundary integral approach independent of boundary conditions, allowing decoupled wave propagation and reflection analysis, extending beyond previous semiclassical methods.

Findings

Exact transfer operator descriptions incorporating diffraction and evanescent effects.

Decoupling of interior wave propagation from boundary reflections.

Application to calculating Goos-Hänchen shifts in complex billiard and cavity systems.

Abstract

We propose a reformulation of the boundary integral equations for the Helmholtz equation in a domain in terms of incoming and outgoing boundary waves. We obtain transfer operator descriptions which are exact and thus incorporate features such as diffraction and evanescent coupling; these effects are absent in the well known semiclassical transfer operators in the sense of Bogomolny. It has long been established that transfer operators are equivalent to the boundary integral approach within semiclassical approximation. Exact treatments have been restricted to specific boundary conditions (such as Dirichlet or Neumann). The approach we propose is independent of the boundary conditions, and in fact allows one to decouple entirely the problem of propagating waves across the interior from the problem of reflecting waves at the boundary. As an application, we show how the decomposition may be used to calculate Goos-H\"anchen shifts of ray dynamics in billiards with variable boundary conditions and for dielectric cavities.