Circular and Elliptical Neutrino Billiards: A Semiclassical Approach

Abstract

Quantum signatures of classical chaos in the spectral properties of non-relativistic quantum billiards are well understood by now based on the semiclassical and the random matrix theory aproach. The experimental detection of relativistic phenomena featured by graphene implicated an increasing interest in the properties of graphene billiards and relativistic quantum billiards consisting of massless spin-1/2 particles governed by the Dirac equation and confined to a planar domain. Confinement is attained by imposing appropriate boundary conditions on the wave function components. We chose those proposed for neutrino billiards. By now numerical studies have been performed on the spectral properties of neutrino billiards of various shapes. Our main focus is the question to what extent the semiclassical approach in terms of trace formulae applies to such systems. We derive trace formulae for circular and elliptical neutrino billiards. The associated eigenvalue equations have been solved analytically for the circular billiard. We present analytical solutions for elliptical neutrino billiards.