SPECTRAL DUALITY FOR PLANAR BILLIARDS

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 Omega(c). We show t hat the on-shell S-matrices S-k have eigenvalues converging to 1 as k up arrow k(0) exactly when -Delta(Omega) has an eigenvalue at energy k (0)(2). This includes multiplicities, and proves a weak form of ''tran sparency'' at k = k(0). We also show that stronger forms of transparen cy, such as S-k0 having an eigenvalue 1 are not expected to hold in ge neral.