Spectral duality for planar billiards

For a bounded open domain Ω with connected complement inR2 and piecewise smooth boundary, we consider the Dirichlet Laplacian-ΔΩ on Ω and the S-matrix on the complementΩc. We show that the on-shell S-matricesSk have eigenvalues converging to 1 ask↑k0 exactly when--ΔΩ has an eigenvalue at energyk02. This includes multiplicities, and proves a weak form of “transparency” atk=k0. We also show that stronger forms of transparency, such asSk0 having an eigenvalue 1 are not expected to hold in general.