Energy level statistics in the transition regime between integrability andchaos for systems without an anti-unitary symmetry

Energy spectra of a particle with mass rn and charge e in the cubic Aharono v-Bohm billiard containing around 10(4) consecutive levels starting from th e ground state have been analysed. The cubic Aharonov-Bohm billiard is a pl ane billiard defined by the cubic conformal mapping of the unit disc pervad ed by a point magnetic flux through the origin perpendicular to the plane o f the billiard. The magnetic Bur does not influence the classical dynamics, but breaks the antiunitary symmetry in the system, which affects the stati stics of energy levels. By varying the shape parameter lambda the classical dynamics goes from integrable (lambda = 0) to fully chaotic (lambda = 0.2; Africa billiard). The level spacing distribution P(S) and the number varia nce Sigma(2)(L) have been studied for 13 different shape parameters on the interval (0 less than or equal to lambda less than or equal to 0.2). Gaussi an unitary ensembles statistics has proven correct for the completely chaot ic case, while in the mixed regime the fractional power-law level repulsion has been observed. The exponent of the level repulsion has been analysed a nd is found to change smoothly from 0 to 2 as the dynamics goes from integr able to ergodic. This is precisely the analogy of the fractional power-law level repulsion observed in the Poisson-Gaussian orthogonal ensemble (GOE) transition by Prosen and Robnik (1993, 1994) and it thus essentially differ s from the prediction of the random matrix theories. The semiclassical Beny -Robnik theory is expected to be correct in the ultimate semiclassical limi t. However, we argue that the semiclassical regime has not been reached and give an estimate for the number of energy levels required for the Beny-Rob nik statistics to apply.