Spectral Duality for Planar Billiards
نویسنده
چکیده
ao -d yn /9 40 50 01 2 M ay 1 99 4 Spectral Duality for Planar Billiards J.-P. Eckmann1;2 and C.-A. Pillet1 Dépt. de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland Section de Mathématiques, Université de Genève, CH-1211 Genève 4, Switzerland Abstract. For a bounded open domain with connected complement in R2 and piecewise smooth boundary, we consider the Dirichlet Laplacian on and the S-matrix on the complement c. We show that the on-shell S-matrices Sk have eigenvalues converging to 1 as k " k0 exactly when has an eigenvalue at energy k2 0 . This includes multiplicities, and proves a weak form of “transparency” at k = k0. We also show that stronger forms of transparency, such as Sk0 having an eigenvalue 1 are not expected to hold in general.
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