Reaction matrix for Dirichlet billiards with attached waveguides

TL;DR

This paper develops a convergent spectral representation of the reaction matrix for Dirichlet billiards with attached waveguides, enabling accurate analysis of scattering properties in systems like rectangular and Sinai billiards.

Contribution

It introduces a new convergent spectral sum for the reaction matrix in Dirichlet billiards, overcoming previous divergence issues and improving scattering analysis.

Findings

Derived a convergent spectral sum for the reaction matrix.

Validated the approach on rectangular and Sinai billiards.

Enhanced understanding of scattering in billiard systems.

Abstract

The reaction matrix of a cavity with attached waveguides connects scattering properties to properties of a corresponding closed billiard for which the waveguides are cut off by straight walls. On the one hand this matrix is directly related to the S-matrix, on the other hand it can be expressed by a spectral sum over all eigenfunctions of the closed system. However, in the physically relevant situation where these eigenfunctions vanish on the impenetrable boundaries of the closed billiard, the spectral sum for the reaction matrix, as it was used before, fails to converge and does not reliably reproduce the scattering properties. We derive here a convergent representation of the reaction matrix in terms of eigenmodes satisfying Dirichlet boundary conditions and demonstrate its validity in the rectangular and the Sinai billiards.