On the Integral Geometry of Liouville Billiard Tables

TL;DR

This paper introduces a Radon transform for integrable billiard tables, proves its injectivity for certain Liouville tables, and establishes spectral rigidity results for associated Laplace-Beltrami operators.

Contribution

It develops a Radon transform framework for Liouville billiard tables and proves spectral rigidity, advancing understanding of integrable billiard dynamics and inverse spectral problems.

Findings

Radon transform is one-to-one for certain Liouville billiard tables

Frequency map is non-degenerate for a class of Liouville tables

Spectral rigidity of Laplace-Beltrami operators with Robin boundary conditions

Abstract

The notion of a Radon transform is introduced for completely integrable billiard tables. In the case of Liouville billiard tables of dimension 3 we prove that the Radon transform is one-to-one on the space of continuous functions $K$ on the boundary which are invariant with respect to the corresponding group of symmetries. We prove also that the frequency map associated with a class of Liouville billiard tables is non-degenerate. This allows us to obtain spectral rigidity of the corresponding Laplace-Beltrami operator with Robin boundary conditions.