Inverse scattering at fixed energy on asymptotically hyperbolic\n Liouville surfaces
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Bibliographic record
Abstract
In this paper, we study an inverse scattering problem on Liouville surfaces\nhaving two asymptotically hyperbolic ends. The main property of Liouville\nsurfaces consists in the complete separability of the Hamilton-Jacobi equations\nfor the geodesic flow. An important related consequence is the fact that the\nstationary wave equation can be separated into a system of a radial and angular\nODEs. The full scattering matrix at fixed energy associated to a scalar wave\nequation on asymptotically hyperbolic Liouville surfaces can be thus simplified\nby considering its restrictions onto the generalized harmonics corresponding to\nthe angular separated ODE. The resulting partial scattering matrices consists\nin a countable set of $2 \\times 2$ matrices whose coefficients are the so\ncalled transmission and reflection coefficients. It is shown that the\nreflection coefficients are nothing but generalized Weyl-Titchmarsh functions\nfor the radial ODE in which the generalized angular momentum is seen as the\nspectral parameter. Using the Complex Angular Momentum method and recent\nresults on 1D inverse problem from generalized Weyl-Titchmarsh functions, we\nshow that the knowledge of the reflection operators at a fixed non zero energy\nis enough to determine uniquely the metric of the asymptotically hyperbolic\nLiouville surface under consideration.\n
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