Spectral Duality for Planar Billiards

Abstract

For a bounded open domain $\Omega$ with connected complement in ${\bf R}^2$ and piecewise smooth boundary, we consider the Dirichlet Laplacian $-\Delta_\Omega$ on $\Omega$ and the S-matrix on the complement $\Omega^c$. We show that the on-shell S-matrices ${\bf S}_k$ have eigenvalues converging to 1 as $k\uparrow k_0$ exactly when $-\Delta_\Omega$ has an eigenvalue at energy $k_0^2$. This includes multiplicities, and proves a weak form of ``transparency'' at $k=k_0$. We also show that stronger forms of transparency, such as ${\bf S}_{k_0}$ having an eigenvalue 1 are not expected to hold in general.