Orbit bifurcations and wavefunction autocorrelations

Abstract

It was recently shown (Keating & Prado, {\it Proc. R. Soc. Lond. A} {\bf 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 $\hbar^{\alpha}$ and a width that scales as $\hbar^{\omega}$, where $\alpha$ and $\omega$ are bifurcation-dependent scar exponents whose values are typically smaller than those ($\alpha=\omega=1/2$) 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 $\hbar^{1-\alpha}$, where $\alpha$ 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.