EXACT THEORY FOR THE QUANTUM EIGENSTATES OF A STRONGLY CHAOTIC SYSTEM

We present an exact theory for the quantum eigenstates of a strongly chaotic system, which consists of a point particle sliding freely on a two-dimensional compact surface of constant negative curvature. The main result is a general sum rule which relates an (almost) arbitrarily smoothed sum over the quantum wavefunctions to a sum over the lengths of the classical orbits. As an example, we apply this formula to the Gaussian smoothed sum over wavefunctions. The question of "scars" as imprints of individual closed orbits is discussed, and the connection with Bogomolny's semiclassical theory is explicitly worked out. The investigation of the amplitude distribution P(PSI) reveals that the higher excited wavefunctions PSI behave as Gaussian random waves.