  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.
