THE CAT MAPS - QUANTUM-MECHANICS AND CLASSICAL MOTION

The relationship between the hyperbolic automorphisms of the unit 2-torus and their quantum counterparts is studied. An exact and explicit representation of the quantum density of states in terms of the periodic orbits of the classical motion is derived. This is then used to obtain exact formulae for certain statistics of the distribution of the eigenvalues of the quantum propagator. It is shown that these statistics are strongly dependent upon the arithmetical nature of the dimensionless Planck's constant. We investigate their semiclassical form and, using results concerning the distribution of the long period orbits, find that they do not correspond to those of any of the universality classes previously studied. Finally, formulae are derived which relate properties of the quantum eigenfunctions to the periodic orbits.