  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.
