The motion of energy levels in quantum systems that show a chaotic classical limit is statistically analyzed. A quantitative comparison is made between the tails of the curvature distribution and numerical results obtained for various physical models. Approximate analytic expressions for the full curvature distribution are derived from the statistical mechanics of a fictitious gas in a refined formulation that recovers the random-matrix theory for an arbitrary number of levels. They provide a better description of numerical data than just the tail-limiting expressions available previously. Good agreement with numerical data for various physical systems as well as for a model random dynamics is obtained with the ad hoc introduced, very simple analytic expressions containing no free parameter. The nonuniversal behavior of small curvatures is discussed. The data obtained for the magnetized hydrogen atom support the previous interpretation of this phenomenon as due to the ‘‘scarred’’ wave functions. A large number of analyzed data allows us to show that the details of the curvature distribution provide a qualitative measure of the degree of scarring in different systems.

• Received 6 October 1992

DOI:https://doi.org/10.1103/PhysRevE.47.1650