Scattering on an hyperbolic torus in a constant magnetic field

We study the quantum mechanical scattering of a particle on a two dimensional non compact Riemannian manifold of constant negative curvature embedded in a constant external magnetic field. The interplay between fundamental group operations and gauge tranformations allows to compute the scattering states. A closed expression for the phase shift of a plane wave entering the torus through its leak is shown to involve the quantized magnetic flux