Spectral Duality for Planar Billiards

Abstract

For a bounded open domain $\Omega$ with connected complement in ${\mathbf{R}}^2$ 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 ${\mathbf{S}}_k$ 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 ${\mathbf{S}}_{k_0}$ having an eigenvalue 1 are not expected to hold in general.

33 pages, Postscript, A4

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