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–Haenchen shifts of ray dynamics in billiards with variable boundary conditions and for dielectric cavities.
Creagh, S. C., Hamdin, H. B., & Tanner, G. (2013). In–out decomposition of boundary integral equations. Journal of Physics A: Mathematical and Theoretical, 46(43), doi:10.1088/1751-8113/46/43/435203