Orbit Bifurcations and the Scarring of Wavefunctions

Keating, J.P.; Prado, S.D.

HPL-BRIMS-2000-21

Keyword(s): bifurcations; scars; quantum chaos

Abstract: Please Note. This abstract contains mathematical formulae which cannot be represented here. We extend the semiclassical theory of scarring of quantum eigenfunctions by classical periodic orbits to include situations where these orbits undergo generic bifurcations. It is shown that , averaged locally with respect to position q and the energy spectrum , has structure around bifurcating periodic orbits with an amplitude and length-scale whose -dependence is determined by the bifurcation in question. Specifically, the amplitude scales as and the length- scale as , and values of the scar exponents, and , are computed for a variety of generic bifurcations. In each case, the scars are semiclassically wider than those associated with isolated and unstable periodic orbits; moreover, their amplitude is at least as large, and in most cases larger. In this sense, bifurcations may be said to give rise to superscars. The competition between the contributions from different bifurcations to determine the moments of the averaged eigenfunction amplitude is analysed. We argue that there is a resulting universal -scaling in the semiclassical asymptotics of these moments for irregular states in systems with a mixed phase-space dynamics. Finally, a number of these predictions are illustrated by numerical computations for a family of perturbed cat maps. Notes: S.D. Prado, Instituto de Fiscia, Universidade Federal do Rio Grande do Sul P.O. Box 15051, 91501-970 Porto Alegre, RS Brazil.

24 Pages

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