Probability distributions in classical and quantum elliptic billiards

Advances in the fabrication of nanocavities to confine electrons have popularized newly the study of 2D quantum wells. The analogies and differences between classical and quantum probability distributions and energy spectra of a particle confined in an elliptic billiard are presented. Classically, the probability densities are characterized by the eigenvalues of an equation that involves elliptic integrals, whereas the ordinary and modified Mathieu functions are applied to describe the quantum distributions. The transition from the elliptic geometry toward the circular geometry is analyzed as well. The problem is interesting itself because it presents strong analogies with the electromagnetic propagation in elliptic waveguides.