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 Latin small letter h with stroke. In the case of the unbroken separatrix, an interesting resonance phenomenon, which occurs for certain values of Latin small letter h with stroke, 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 TlnLatin small letter h with stroke. The implications of these results to other systems is also discussed.