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.