1. Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces
- Author
-
François Nicoleau, Thierry Daudé, Niky Kamran, Analyse, Géométrie et Modélisation (AGM - UMR 8088), CY Cergy Paris Université (CY)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), and French National Research Projects AARG, No. ANR-12-BS01-012-01, and Iproblems, No. ANR-13-JS01-0006 and by the UMI centre de recherches mathématiques 3457} * NSERC grant RGPIN 105490-2011} * French National Research Project NOSEVOL, No. ANR- 2011 BS0101901}
- Subjects
Angular momentum ,Inverse scattering ,FOS: Physical sciences ,Liouville surface ,Weyl-Titchmarsh function ,01 natural sciences ,Singular Sturm-Liouville problems ,Theoretical Computer Science ,Matrix (mathematics) ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,0101 mathematics ,[MATH]Mathematics [math] ,Mathematical Physics ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Ode ,Mathematical Physics (math-ph) ,Inverse problem ,Wave equation ,Computer Science Applications ,010101 applied mathematics ,Reflection (mathematics) ,[CHIM.POLY]Chemical Sciences/Polymers ,Signal Processing ,Inverse scattering problem ,81U40, 35P25, 58J50 - Abstract
In this paper, we study an inverse scattering problem on Liouville surfaces having two asymptotically hyperbolic ends. The main property of Liouville surfaces consists in the complete separability of the Hamilton-Jacobi equations for the geodesic flow. An important related consequence is the fact that the stationary wave equation can be separated into a system of a radial and angular ODEs. The full scattering matrix at fixed energy associated to a scalar wave equation on asymptotically hyperbolic Liouville surfaces can be thus simplified by considering its restrictions onto the generalized harmonics corresponding to the angular separated ODE. The resulting partial scattering matrices consists in a countable set of $2 \times 2$ matrices whose coefficients are the so called transmission and reflection coefficients. It is shown that the reflection coefficients are nothing but generalized Weyl-Titchmarsh functions for the radial ODE in which the generalized angular momentum is seen as the spectral parameter. Using the Complex Angular Momentum method and recent results on 1D inverse problem from generalized Weyl-Titchmarsh functions, we show that the knowledge of the reflection operators at a fixed non zero energy is enough to determine uniquely the metric of the asymptotically hyperbolic Liouville surface under consideration., 39 pp
- Published
- 2015