STATISTICAL PROPERTIES OF HIGHLY EXCITED QUANTUM EIGENSTATES OF A STRONGLY CHAOTIC SYSTEM

Statistical properties of highly excited quantal eigenstates are studi ed for the free motion (geodesic flow) on a compact surface of constan t negative curvature (hyperbolic octagon) which represents a strongly chaotic system (K-system). The eigenstates are expanded in a circular- wave basis, and it turns out that the expansion coefficients behave as Gaussian pseudo-random numbers. It is shown that this property leads to a Gaussian amplitude distribution P(PSI) in the semiclassical limit , i.e. the wave-functions behave as Gaussian random functions. This be haviour, which should hold for chaotic systems in general, is nicely c onfirmed for eigenstates lying 10 000 states above the ground state th us probing the semiclassical limit. In addition, the autocorrelation f unction and the path-correlation function are calculated and compared with a crude semiclassical Bessel-function approximation. Agreement wi th the semiclassical prediction is only found, if a local averaging is performed over roughly 1000 de Broglie wavelengths. On smaller scales , the eigenstates show much more structure than predicted by the first semiclassical approximation.