Abstract
For a bounded open domain Ω with connected complement inR 2 and piecewise smooth boundary, we consider the Dirichlet Laplacian-Δ Ω on Ω and the S-matrix on the complementΩ c. We show that the on-shell S-matricesS k have eigenvalues converging to 1 ask↑k 0 exactly when--Δ Ω has an eigenvalue at energyk 20 . This includes multiplicities, and proves a weak form of “transparency” atk=k 0. We also show that stronger forms of transparency, such asS k 0 having an eigenvalue 1 are not expected to hold in general.
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Communicated by A. Kupiainen
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Eckmann, JP., Pillet, CA. Spectral duality for planar billiards. Commun.Math. Phys. 170, 283–313 (1995). https://doi.org/10.1007/BF02108330
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DOI: https://doi.org/10.1007/BF02108330