Statistical Modeling of Wave Chaotic Transport and Tunneling

Abstract

This thesis treats two general problem areas in the field of wave chaos. The first problem area that we address concerns short wavelength tunneling from a classically confined region in which the classical orbits are chaotic. We de- velop a quantitative theory for the statistics of energy level splittings for symmetric chaotic wells separated by a tunneling barrier. Our theory is based on the ran- dom plane wave hypothesis. While the fluctuation statistics are very different for chaotic and non-chaotic well dynamics, we show that the mean splittings of differ- ently shaped wells, including integrable and chaotic wells, are the same if their well areas and barrier parameters are the same. We also consider the case of tunneling from a single well into a region with outgoing quantum waves. Our second problem area concerns the statistical properties of the impedance matrix (related to the scattering matrix) describing the input/output properties of waves in cavities in which ray trajectories that are regular and chaotic coexist (i.e., `mixed' systems). The impedance can be written as a summation over eigenmodes where the eigenmodes can typically be classified as either regular or chaotic. By appropriate characterizations of regular and chaotic contributions, we obtain statis- tical predictions for the impedance. We then test these predictions by comparison with numerical calculations for a specific cavity shape, obtaining good agreement.