Transfer operator approach on three-dimensional quantum billiards with SO(2) symmetry.

This work demonstrates the application of Bogomolny's transfer operator method on three-dimensional dynamics. Motivated by experimental observations of lenslike metal clusters, the quantum billiards bounded by a flat bottom and an upper surface with SO(2) symmetry are studied. A precise determination of the energies with error less than 0.05% and exact predicted degeneracies in the special case of the half-sphere billiard confirm the efficiency of this method. Furthermore, the spectra and degeneracies of lens billiards with varying heights are explicitly determined.