02-182 Georgi Popov, Petar Topalov

Liouville billiard tables and an inverse spectral result (83K, LaTeX 2e) Apr 11, 02

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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 $G$ .

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