Georgi Popov, Peter Topalov
Invariants of isospectral deformations and spectral rigidity
(423K, pdf)
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.