Solutions of the Lippmann-Schwinger equation for confocal parabolic billiards

We present analytical and numerical solutions of the Lippmann-Schwinger equation for the scattered wave functions generated by confocal parabolic billiards and parabolic segments with various 3-type potential-strength functions. The analytical expressions are expressed as summations of products of parabolic cylinder functions Dm. We numerically investigate the resonances and tunneling in the confocal parabolic billiards by employing an accurate boundary wall method that provides a complete inside-outside picture. The criterion for discretizing the parabolic sides of the billiard is explained in detail. We discuss the phenomenon of transparency at certain eigenenergies.