  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.
