 94118 Eckmann J.P., Pillet C.A.
 Spectral Duality for Planar Billiards
(432K, Postscript)
May 2, 94

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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 Smatrix on the complement $\Omega^c$.
We show that the onshell Smatrices ${\bf S}_k$ 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.
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