STATISTICAL PROPERTIES OF PARAMETER-DEPENDENT CLASSICALLY CHAOTIC QUANTUM-SYSTEMS

In this paper we examine the dependence of the energy levels of a classically chaotic system on a parameter. We present numerical results which justify the use of a random matrix model for the statistical properties of this dependence. We illustrate the application of our model by calculating both the number of avoided crossings as a function of gap size and the distribution of curvatures of energy levels for a chaotic billiard: the distribution of large curvatures is determined by the density of avoided crossings. Our results confirm that the matrix elements are Gaussian distributed in the semiclassical limit, but we characterize significant deviations from the Gaussian distribution at finite energies.