  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 $\delta$-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.
