Relativistic Quantum Billiards with Threefold Rotational Symmetry: Exact, Symmetry-Projected Solutions for Equilateral Neutrino Billiard

B. Dietz Lanzhou Center for Theoretical Physics and the Gansu Provincial Key Laboratory of Theoretical Physics, Lanzhou University, Lanzhou, Gansu 730000, China

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We present analytical results for the eigenvalues and eigenfunctions of a relativistic neutrino billiard with the shape of an equilateral triangle, which are valid from the ultrarelativistic to the nonrelativistic limit. The transition is performed by increasing the mass of the neutrino from zero to infinity. Here, we exploit the threefold symmetry of the triangle to separate the eigenstates according to their transformation properties with respect to rotation by 2π/3 into three subspaces labeled by l=0,1,2. Generally, the boundary condition imposed on the spinor eigenfunctions of the Dirac Hamiltonian of a neutrino billiard with threefold symmetry, in order to confine the neutrino to the billiard domain, yields a relation of the first spinor component to the second one, where both belong to distinct irreducible representations. In order to obtain information on the effect of this entanglement of different symmetry classes on the eigenstates, we investigate the spectral properties and properties of the eigenfunctions for different masses and the transition from the ultrarelativistic to the nonrelativistic limit.

DOI:10.12693/APhysPolA.140.473 topics: relativistic quantum billiards, semiclassical theory, discrete symmetries