Quantizing neutrino billiards: an expanded boundary integral method

With the pioneering fabrication of graphene the field of relativistic quantum chaos emerged. We will focus on the spectral properties of massless spin-1/2 particles confined in a bounded two-dimensional region, named neutrino billiards by Berry and Mondragon in 1987. A commonly used method for the determination of the eigenvalues is based on a boundary integral equation originating from Green’s theorem. Yet, in the quantization one might face problems similar to those occurring for non-relativistic quantum billiards. Especially in cases where the eigenvalue spectrum contains near degeneracies the identification of complete sequences of eigenvalues might be extremely elaborate, if not unfeasible. We propose an expanded boundary integral method, which yields complete eigenvalue sequences with a considerably lower numerical effort than the standard one. Actually, it corresponds to an extension of the method introduced in Veble et al (2007 New J. Phys. 9 15) to relativistic quantum billiards. To demonstrate its validity and its superior efficiency compared to the standard method, we apply both methods to a circular billiard of which the eigenvalues are known analytically and exhibit near degeneracies. Finally, we employ it for the quantization of a neutrino billiard with a hole, of which the spectrum contains many close lying levels and exhibits unusual fluctuation properties.