Solutions of the Lippmann-Schwinger equation for confocal parabolic billiards
We present analytical and numerical solutions of the Lippmann-Schwinger equation for the scattered wavefunctions generated by confocal parabolic billiards and parabolic segments with various $δ$-type potential-strength functions. The analytical expressions are expressed as summations of products of parabolic cylinder functions $D_m$. 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.
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