- Orbit bifurcations and wavefunction autocorrelations
- A. Bäcker, J. P. Keating and S. Prado
Nonlinearity 15 (2002) 1417-1433
- It was recently shown (Keating & Prado, Proc. R. Soc. Lond.
A 457, 1855-1872 (2001))
that, in the semiclassical limit,
the scarring of quantum eigenfunctions by classical periodic orbits
in chaotic systems may be
dramatically enhanced when the orbits in question undergo
bifurcation. Specifically, a bifurcating orbit gives rise to a
scar with an amplitude that scales as
and a width that scales as
where α and ω
are bifurcation-dependent scar exponents whose values are
typically smaller than those (α=ω=1/2) associated with
isolated and unstable periodic orbits. We here analyze the
influence of bifurcations on the autocorrelation function of
quantum eigenstates, averaged with respect to energy. It is shown
that the length-scale of the correlations around a bifurcating orbit
scales semiclassically as h1-α, where α is the
scar amplitude exponent. This imprint of bifurcations on quantum
autocorrelations is illustrated by numerical computations for a
family of perturbed cat maps.
- Please refer to the above published version.
(The preprint version of this article can be obtained as
Ulm report ULM-TP/02-2
(April 2002), 25pp.
Bifurcations of periodic orbits may have a substantial
influence on the autocorrelation function.
This is illustrated in the following figure
where for the perturbed cat maps
the (absolute value of the) autocorrelation function c(l)
is shown as a function of the perturbation parameter κ.
At bifurcations of periodic orbits and l
strongly enhanced peaks are observed.
Last modified: 27 October 2004, 16:21:49
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