Quantum properties of irrational triangular billiards

Abstract

Triangles with sides given by consecutive integers (N , N+1 , N+2) are fully irrational (all angles irrational with pi) if 3<N<infinity . Rational approximations to their angles and the Hurwitz theorem in number theory are used to define a parameter h that quantifies the irrationality of each triangle. The energy level statistics [spacing distribution p(s) and spectral rigidity Delta(3)(L)] of quantum billiards from this one-parameter family of triangles are investigated. The behavior of h with varying N and the numerically calculated level dynamics are found to be closely related: h exhibits a local maximum at N=10, around which agreement with Gaussian orthogonal ensemble (GOE) spectral fluctuations is observed. As N is increased, h decreases and the statistics depart from GOE. Structures appear in p(s) for N>120 and eventually the occurrence of gaps in the distribution for N ~ 180 define the onset of a long crossover towards the sequence observed in the integrable limit of the equilateral triangle (N-->infinity) .