Parametric motion of energy levels: Curvature distribution

We study the statistical properties of the distortions of irregular energy spectra when a perturbation parameter is varied; for example, the strength of an external field acting on the bounded quantum system. Three kinds of generalized Calogero-Moser (GCM) classical Hamiltonians are shown to rule the parametric motion of the energy levels in orthogonal, unitary, and symplectic systems. Using these GCM Hamiltonians, we construct a Newtonian theory of ensembles where irregular spectra are correlated with the properties of infinite gases of GCM particles. In this dynamical approach, the results of random matrix theory are recovered. Furthermore, we are able to study parametric properties of irregular spectra such as the level curvature defined by the second derivative of a level energy with respect to the perturbation parameter. We prove that the level curvature density of the orthogonal, unitary, and symplectic systems decreases, respectively, as {}K^-3, {}K^-4, and {}K^-6 for large curvature {}K{}. We present numerical results supporting our theoretical analysis and suggesting the universality of the curvature distribution. The relationship of the curvature distribution to the spacing distribution, as well as the possible experimental observation of the curvature distribution, is discussed.