Semi-Poisson Statistics in Relativistic Quantum Billiards with Shapes of Rectangles

Rectangular billiards have two mirror symmetries with respect to perpendicular axes and a twofold (fourfold) rotational symmetry for differing (equal) side lengths. The eigenstates of rectangular neutrino billiards (NBs), which consist of a spin-1/2 particle confined through boundary conditions to a planar domain, can be classified according to their transformation properties under rotation by p (p/2) but not under reflection at mirror-symmetry axes. We analyze the properties of these symmetry-projected eigenstates and of the corresponding symmetry-reduced NBs which are obtained by cutting them along their diagonal, yielding right-triangle NBs. Independently of the ratio of their side lengths, the spectral properties of the symmetry-projected eigenstates of the rectangular NBs follow semi-Poisson statistics, whereas those of the complete eigenvalue sequence exhibit Poissonian statistics. Thus, in distinction to their nonrelativistic counterpart, they behave like typical quantum systems with an integrable classical limit whose eigenstates are non-degenerate and have alternating symmetry properties with increasing state number. In addition, we found out that for right triangles which exhibit semi-Poisson statistics in the nonrelativistic limit, the spectral properties of the corresponding ultrarelativistic NB follow quarter-Poisson statistics. Furthermore, we analyzed wave-function properties and discovered for the right-triangle NBs the same scarred wave functions as for the nonrelativistic ones.

Automated summary

Rectangular billiards have mirror symmetries and rotational symmetries, and the eigenstates can be classified based on their transformation properties. The properties of these symmetry-projected eigenstates and symmetry-reduced billiards are analyzed. The spectral properties of the eigenstates of the rectangular billiards follow semi-Poisson statistics, while the complete eigenvalue sequence exhibits Poissonian statistics. Quarter-Poisson statistics are found for the ultrarelativistic billiards corresponding to right triangles with semi-Poisson statistics in the nonrelativistic limit. Scarred wave functions are discovered for the right-triangle billiards.