QUANTUM SIGNATURES OF HOMOCLINIC TANGLES AND SEPARATRICES

The influence of homoclinic tangles and separatrices on the eigenstates and quasienergies of a quantized, nonintegrable, piecewise-linear standard map is investigated both analytically and numerically. The strength of scarring on hyperbolic fixed points is found to depend strongly on the phase-space area covered by the corresponding homoclinic tangles. It is particularly pronounced when the tangle degenerates to an unbroken separatrix and is seen to decrease with decreasing hBAR. In the case of the unbroken separatrix, an interesting resonance phenomenon, which occurs for certain values of hBAR, is also discussed. This effect is related to a degeneracy in the quasienergy differences for a triplet of states which lie energetically below, on, and above the saddle. This degeneracy is shown to give rise to periodic recurrences for a coherent state initially placed at the hyperbolic fixed point. The relevant time scale T is calculated semiclassically and exhibits the asymptotic behavior T is-proportional-to ln hBAR. The implications of these results to other systems is also discussed.