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The statistical fluctuations of the energy levels of classically chaotic systems are believed to coincide with those of Gaussian matrix en$embles. The tramition from the Gaussian Onhogonal Ensemble (GOE) to lhe Gaussian Unitary Ensemble (GUE). as the timereversal symmetry is broken. is known exactly. The asymptotic form ofthe two-point correlation function is here derived from semiclassical periodic orbit theory, leading to a dynamical evaluation of the uansition parameter. Numerical calculations of the correlation function for a chaotic billiard in B constant magnetic field reveal a clear c~ossover from GUE behaviour to GOE as the level separation is increased, in agreement with the theoretical prediction.

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2014 Nous étudions les propriétés de fluctuation des valeurs propres du Laplacien à deux dimensions avec des conditions aux limites de Dirichlet sur un stade. Elles sont consistantes avec les fluctuations des valeurs propres de matrices aléatoires (GOE). Nous faisons la conjecture que ceci est vrai en général, pourvu que la frontière soit telle que le mouvement d'une particule libre réfléchie élastiquement par la frontière (billard) soit un mouvement très chaotique. Abstract. 2014 We investigate the fluctuation properties of the eigenvalues of the Laplacian in two dimensions with Dirichlet boundary conditions on a stadium. They are found to be consistent with the fluctuations of eigenvalues of random matrices (GOE). It is conjectured that this is true for any boundary such that the motion of a free particle elastically reflected by the boundary is a strongly chaotic motion.