Two-dimensional quantum scattering by non-isotropic interactions localized on a circle, applications to open billiards

Two-dimensional quantum scattering by isotropic and non-isotropic interactions localized on a circle is considered. The expansion of the interaction on the circle in a Fourier series allows us to express basic objects of scattering theory (resolvent, T operator, differential cross length, cross length, and cross length averaged over all orientations of the incident particle), in terms of operations on matrices. For numerical applications, these matrices are truncated to a given order. If the interaction is isotropic, this general formulation reduces to the usual one, and the resonances in the isotropic cases are studied because they allow us to interpret resonances in some non-isotropic cases. Applications to open circular billiards are given. A first approach to the open quantum circular billiard is an interaction equal to zero on some parts of the circle and to a great value lambda on the other parts. Then, it turns out that the limit lambda -> + infinity does not lead to an explicit result in this approach. A second approach to the open circular billiard is given by replacing apertures (where the interaction is zero) by pseudo-apertures (where the interaction is small but non-zero), and the limit lambda -> + infinity then leads to an explicit result. In the high energy regime, the two approaches give similar results and allow interpretation of numerous results, in particular, for differential cross lengths, in terms of classical mechanics. Published by AIP Publishing.