Spectral correlation functions for chaotic systems

The n-level spectral correlation functions for chaotic quantum systems are calculated by using a recently proposed scheme called the extended diagonal approximation (EDA). The EDA is a natural extension of the diagonal approximation, which was invented by Berry in order to semiclassically evaluate the 2-level correlation function. When the time reversal invariance of the chaotic systems is broken, the EDA yields n × n determinant expressions for the n-level correlation functions, which exactly agree with the predictions of the random matrix theory. On the other hand, when the system is time reversal invariant, only the leading terms of the random matrix predictions are reproduced. § 1. The diagonal approximation for the spectral correlation function Let us consider a two-dimensional bounded quantum system which is chaotic in the classical limit. The time-reversal invariance is supposed to be broken. We are interested in the distribution of the energy levels Ej , which are the eigenvalues of the system Hamiltonian H. The semiclassical theory describes the limit ~ → 0 and shows that the energy level density (1.1) ρ(E) = ∑