2009/06/02 by Georgi Popov, Popov, G., Peter Topalov +1 · 1 citation
Physics and Astronomy · Mathematics · #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.0906.0449
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 Λ of the billiard ball map with a vector of rotation satisfying a Diophantine condition we prove that certain integrals on Λ 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 Λ. 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.