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On the Integral Geometry of Liouville Billiard Tables

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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.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Jean Leray, CNRS: UMR6629, Université de Nantes, 2, rue de la Houssinière, BP 92208, 44072, Nantes Cedex 03, France

    G. Popov

  2. Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA, 02115, USA

    P. Topalov

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  1. G. Popov
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  2. P. Topalov
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Correspondence to P. Topalov.

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Communicated by S. Zelditch

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Popov, G., Topalov, P. On the Integral Geometry of Liouville Billiard Tables. Commun. Math. Phys. 303, 721–759 (2011). https://doi.org/10.1007/s00220-011-1223-z

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  • DOI: https://doi.org/10.1007/s00220-011-1223-z

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