05-62 Georgi Popov, Peter Topalov

Invariants of isospectral deformations and spectral rigidity (423K, pdf) Feb 10, 05

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Abstract. We introduce a notion of weak isospectrality for continuous deformations. Let us consider the Laplace-Beltrami operator on a compact Riemannian manifold with boundary with Robin boundary conditions. Given a Kronecker invariant torus of the billiard ball map with a Diophantine vector of rotation we prove that certain integrals on it involving the function in the Robin boundary conditions remain constant under weak isospectral deformations. To this end we construct continuous families of quasimodes. We obtain also isospectral invariants of the Laplacian with a real-valued potential on a compact manifold for continuous deformations of the potential. As an application we prove spectral rigidity in the case of Liouville billiard tables of dimension two.

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