Spectral duality for planar billiards
Résumé
For a bounded open domain Omega with connected complement in R^2 and piecewise smooth boundary, we consider the Dirichlet Laplacian Delta_Omega on Omega and the S-matrix on the complement R^2\Omega. We show that the on-shell S-matrices S(k) have eigenvalues converging to 1 as k -> ko exactly when Delta_Omega has an eigenvalue at energy ko^2. This includes multiplicities, and proves a weak form of "transparency" at k = ko. We also show that stronger forms of transparency, such as S(k) having an eigenvalue 1 are not expected to hold in general.