Generalized Quantum Eigenvalue Formulation for the Neutrino Billiard via Boundary Integral Method

Massless neutrinos are spin-1/2 fermions governed by the relativistic Dirac equation. By exploiting series expansions of the Bessel functions that separately represent the real and imaginary parts of the free-space Green’s function of the (2+1)-dimensional massless Dirac equation, it can be shown that the boundary integral equation for the particle’s wave function inside a neutrino billiard becomes independent of the wave numbers. Consequently, the spectral problem reduces to a generalized eigenvalue problem that can be solved directly for the discrete energy eigenvalues, which are essential for studying the distribution of the system’s energy levels. A numerical example using a circular neutrino billiard with a known analytical solution was used to validate the proposed method, highlighting its potential for broader applications in computational physics and engineering.