Georgi Popov, Petar Topalov
Liouville billiard tables and an inverse spectral result
(83K, LaTeX 2e)
ABSTRACT. We consider a class of billiard tables $(X,g)$, where $X$ is a smooth
compact manifold of dimension 2 with smooth boundary $\partial X$
and $g$ is a
smooth Riemannian metric on $X$, the billiard
flow of which is completely integrable.
The billiard table $(X,g)$ is defined by means of a special double
cover with two branched points and it admits a group of isometries
$G \cong {\bf Z}_2 \times{\bf Z}_2$. Its boundary can be
characterized by the string property, namely, the sum of distances
from any point of $\partial X$ to the branched points is constant.
We provide examples of such
billiard tables in the plane (elliptical regions), on the sphere ${\bf
S}^2$, on the hyperbolic space ${\bf H}^2$, and on quadrics.
The main result is that the spectrum of the corresponding
Laplace-Beltrami operator with Robin boundary conditions involving
a smooth function $K$ on $\partial X$ determines uniquely the function
$K$ provided that $K$ is
invariant under the action of $G$ .