A Bäcker et al 2002 Nonlinearity 15 1417 doi:10.1088/0951-7715/15/5/304
Orbit bifurcations and wavefunction autocorrelations
A Bäcker1, J P Keating2 and S D Prado3
Show affiliationsRecommended by T Prosen
It was recently shown (Keating J P and Prado S D 2001 Proc. R. Soc. A 457 1855–72) 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 
α and a width that scales as ω, where α and ω are bifurcation-dependent scar exponents whose values are typically smaller than those (α = ω = ½) associated with isolated and unstable periodic orbits. We here analyse 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 1−α, where α is the corresponding scar amplitude exponent. This imprint of bifurcations on quantum autocorrelations is illustrated by numerical computations for a family of perturbed cat maps.
- PACS
-
05.45.Mt Quantum chaos; semiclassical methods
- MSC
-
81Q50 Quantum chaos (See also 37Dxx)
37M20 Computational methods for bifurcation problems
81Q20 Semiclassical techniques including WKB and Maslov methods
- Subjects
- Dates
-
Issue 5 (September 2002)
Received 5 April 2002, in final form 18 June 2002
Published 15 July 2002
-
Orbit bifurcations and wavefunction autocorrelations
A Bäcker et al 2002 Nonlinearity 15 1417
-
Symmetries and reversing symmetries of toral automorphisms
Michael Baake and John A G Roberts 2001 Nonlinearity 14 R1
-
About ergodicity in the family of limaçon billiards
Holger R Dullin and Arnd Bäcker 2001 Nonlinearity 14 1673
-
Generic twistless bifurcations
H R Dullin et al 2000 Nonlinearity 13 203
-
The baby Skyrme models and their multi-skyrmions
Tom Weidig 1999 Nonlinearity 12 1489
-
Computing connectedness: An exercise in computational topology
V Robins et al 1998 Nonlinearity 11 913
-
Generating partitions for two-dimensional hyperbolic maps
A Bäcker and N Chernov 1998 Nonlinearity 11 79
-
Quadratic volume-preserving maps
Héctor E Lomelí and James D Meiss 1998 Nonlinearity 11 557
-
How projections affect the dimension spectrum of fractal measures
Brian R Hunt and Vadim Yu Kaloshin 1997 Nonlinearity 10 1031
-
A silenced air pressure control valve
P Bradshaw 1964 J. Sci. Instrum. 41 692
Related review articles
What's this?- The nonlinear Schrödinger equation with a random potential: results and puzzles
- Spectral problems in open quantum chaos
- Tibetan singing bowls