Quantum phase-space structures in classically mixed systems: a scattering approach

Abstract

The structure of quantum wavefunctions is discussed for a billiard system with mixed classical dynamics. Using a scattering formalism, we introduce a Husimi-like distribution for S-matrix eigenvectors at arbitrary wavenumbers, k (not necessarily eigenenergies of the system). This constitutes probability density plots on the Poincare surface of section at any k. Many eigenvectors are structured by classical objects such as invariant tori, the intermediate layer between regular and chaotic motion, and unstable periodic orbits (`scars'). We find that eigenvector structure is stable over significant energy ranges, while the associated eigenphases increase linearly with k at a rate determined by the underlying classical objects. Additionally, we observe that in the presence of time-reversal symmetry, eigenvectors scarred by non-self-retracing periodic orbits form doublets which are closely spaced in eigenphase. Our results provide a new perspective on the quantization of structured states: quantization is determined by the linearly increasing eigenphases, and occurs whenever an eigenphase is equal to , while structure is encoded in the eigenvectors; it is present at all energies and varies only slowly. This allows one to predict many quantized eigenfunctions over an entire energy range by studying the eigenparameters of at a single intermediate energy, and identifying the underlying classical objects.