Orbit bifurcations and the scarring of wave functions

We extend the semiclassical theory of scarring of quantum eigenfunctions Psi (n)(q) by classical periodic orbits to include situations where these orbits undergo generic bifurcations. It is shown that /Psi (n)(q)/(2), averaged locally with respect to position q and the energy spectrum {E(n)} has structure around bifurcating periodic orbits with an amplitude and length-scale whose (h) over bar dependence is determined by the bifurcation in question. Specifically, the amplitude scales as (h) over bar (alpha) and the length-scale as (h) over bar (omega), and values of the scar exponents, alpha and omega, 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 (h) over bar scaling in the semiclassical asymptotics of these moments for irregular states in systems with mixed phase-space dynamics. Finally, a number of these predictions are illustrated by numerical computations for a family of perturbed cat maps.