Invariants of isospectral deformations and spectral rigidity

TL;DR

This paper introduces a new notion of weak isospectrality for continuous deformations of Laplace operators on manifolds, identifying invariants that remain constant under such deformations, and applies this to prove spectral rigidity in specific billiard tables.

Contribution

It defines weak isospectrality for continuous deformations and constructs invariants that remain constant, advancing understanding of spectral rigidity in geometric settings.

Findings

Certain integrals on invariant tori are constant under weak isospectral deformations

Constructed continuous families of quasimodes associated with invariant tori

Proved spectral rigidity for Liouville billiard tables of dimension two

Abstract

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.