Invariants of isospectral deformations and spectral rigidity

Details Invariants of isospectral deformations and spectral rigidity Invariants of isospectral deformations and spectral rigidity

2009-06-02

arxiv.org/abs/0906.0449v3

Mathematics

Spectral Theory

Scientific paper

We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace-Beltrami operator on a compact Riemannian manifold with boundary with Robin boundary conditions. Given a Kronecker invariant torus $\Lambda$ of the billiard ball map with a vector of rotation satisfying a Diophantine condition we prove that certain integrals on $\Lambda$ involving the function in the Robin boundary conditions remain constant under weak isospectral deformations. To this end we construct continuous families of quasimodes associated with $\Lambda$. 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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