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 $G$ .