Spectral Duality for Planar Billiards

For a bounded open domain $\mathrm{\Omega }$ with connected complement in ${\mathbf{R}}^2$ and piecewise smooth boundary, we consider the Dirichlet Laplacian $-{\mathrm{\Delta }}_{\mathrm{\Omega }}$ on $\mathrm{\Omega }$ and the S-matrix on the complement ${\mathrm{\Omega }}^c$. We show that the on-shell S-matrices ${\mathbf{S}}_k$ have eigenvalues converging to 1 as $k\uparrow k_0$ exactly when $-{\mathrm{\Delta }}_{\mathrm{\Omega }}$ 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 ${\mathbf{S}}_{k_0}$ having an eigenvalue 1 are not expected to hold in general.