CHAOTIC DYNAMICS AND THE GOE-GUE TRANSITION

The statistical fluctuations of the energy levels of classically chaot ic systems are believed to coincide with those of Gaussian matrix ense mbles. The transition from the Gaussian Orthogonal Ensemble (GOE) to t he Gaussian Unitary Ensemble (GUE), as the time-reversal symmetry is b roken, is known exactly. The asymptotic form of the two-point correlat ion function is here derived from semiclassical periodic orbit theory, leading to a dynamical evaluation of the transition parameter. Numeri cal calculations of the correlation function for a chaotic billiard in a constant magnetic field reveal a clear crossover from GUE behaviour to GOE as the level separation is increased, in agreement with the th eoretical prediction.