Exact solution to the Lippmann-Schwinger equation for an elliptical billiard

We present an exact solution to the Lippmann-Schwinger equation for a two-dimensional elliptic billiard modeled as a boundary-wall. Using a property of the Green function - which is a zeroth order Hankel function of the first kind - we managed to separate the integral equation Kernel. This approach allows us to study the problem analytically for four representative potential coupling strength gamma(s), namely: (i) gamma(s) = gamma(o)/root cosh(2)mu(0) - cos(2)s; (ii) gamma(s) = gamma(o); (iii) gamma(s) being a 2 pi - periodic even function; and (iv) gamma(s) being a 2 pi - periodic odd function, where gamma(o) is a real constant. Also, we find the condition which needs to be verified in order to the incident plane wave to produce a resonance inside the barrier.