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–Hänchen shifts of ray dynamics in billiards with variable boundary conditions and for dielectric cavities.
Highlights
The transfer operator formalism proposed by Bogomolny [1] has offered a very powerful platform for the application of semiclassical methods to complex quantum and wave problems
In this paper we propose that a decomposition of the wavefunction at the boundary into incoming and outgoing components, defined in detail following a separation of the Green function into its regular and singular parts, provides a natural means of establishing this connection beyond leading semiclassical approximations
We have proposed an approach for separating the boundary solutions of cavity problems into incoming and outgoing components that allows boundary integral equations to be recast as transfer operator equations
Summary
The transfer operator formalism proposed by Bogomolny [1] has offered a very powerful platform for the application of semiclassical methods to complex quantum and wave problems. In this paper we propose that a decomposition of the wavefunction at the boundary into incoming and outgoing components, defined in detail following a separation of the Green function into its regular and singular parts, provides a natural means of establishing this connection beyond leading semiclassical approximations. The main technical issue to be addressed in this paper is how to decompose the wave solution at the boundary into a component approaching the boundary and a component leaving it (see figure 1) This decomposition is automatically achieved in the context of semiclassical approximation when the wavefunction takes the form of an eikonal ansatz, where ray direction allows us to select incoming and reflected wave components. The use of the decomposition of Green operators into local and nonlocal parts to define incoming and outgoing waves is illustrated in section 3 for the one-dimensional case, where calculations are elementary This is generalized to two-dimensional cavities in section 4 using the primitive decomposition.
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