The impact of the classical dynamic on the fluctuation properties in the eigenvalue spectrum of nonrelativistic quantum billiards (QBs) are now well understood based on the semiclassical approach which provides an approximation for the fluctuating part ${\rho }^{\mathrm{fluc}}\left(k\right)$ of the spectral density in terms of a trace formula, that is, a sum over classical periodic orbits of its classical counterpart, abbreviated as CB. This connection between the eigenvalue spectrum of a quantum system and the classical periodic orbits is discernible in the Fourier transform of ${\rho }^{\mathrm{fluc}}\left(k\right)$ from eigenwave number $k$ to length, which exhibits peaks at the lengths of the periodic orbits. The uprise of interest in properties of graphene related to their relativistic Dirac spectrum implicated the emergence of intensive studies of relativistic neutrino billiards (NBs), consisting of a spin-1/2 particle governed by the Dirac equation and confined to a bounded planar domain. In distinction to QBs, NBs do not have a well-defined classical limit. Yet comparison of their length spectra showed that for massless spin-1/2 particles those of the NB exhibit peaks at positions corresponding to the lengths of periodic orbits with an even number of reflections at the boundary of the CB associated with the corresponding QB. In order to understand the transition from the relativistic to the nonrelativistic regime, we derive an exact quantization condition for massive NBs and use it to obtain a trace formula. This trace formula provides a direct link between the spectral density of a NB and the classical dynamic of the corresponding QB through the periodic orbits of the associated CB.

• Received 30 April 2020
• Accepted 30 September 2020

DOI:https://doi.org/10.1103/PhysRevE.102.042214