Orbit bifurcations and spectral statistics

Systems whose phase space is mixed have been conjectured to exhibit quantum spectral correlations that are, in the semiclassical limit, a combination of Poisson and randommatrix, with relative weightings determined by the corresponding measures of regular and chaotic orbits. We here identify an additional component in long-range spectral statistics, associated with periodic orbit bifurcations, which can be semiclassically large. This is illustrated for a family of perturbed cat maps. It has been conjectured that in the semiclassical limit the quantum spectral statistics of classically integrable systems are generically Poissonian, and that those of classically chaotic systems are generically given by the average over an appropriate random-matrix ensemble [1–4]. In between these two extremes lie systems whose phase space is mixed; that is, in which regular and irregular motion coexist. Such systems are said to exhibit soft chaos [5]. For these, it has been suggested that the quantum spectral statistics are a combination of Poisson and random-matrix, with relative weightings determined by the corresponding measures of the regular and chaotic orbits [6]. Our purpose here is to identify in this case an additional component in the long-range statistics that is associated with periodic orbit bifurcations and which can be semiclassically large. A semiclassical theory for long-range spectral statistics has been developed [1, 7–9] based on Gutzwiller’s trace formula [10], which relates the quantum density of states d(E) =∑n δ(E − En) to classical periodic orbits: d(E) = d̄(E)+ 1 πh̄ ∑