Advisory Committee Chair
Gunter Stolz
Document Type
Dissertation
Date of Award
2007
Degree Name by School
Doctor of Philosophy (PhD) College of Arts and Sciences
Abstract
We study a unitary version of the one-dimensional Anderson model, given by a five diagonal deterministic unitary operator multiplicatively perturbed by a random phase matrix. This operator models the time evolution of an electron in a one-dimensional metal ring subject to a magnetic field that linearly increases with time. We fully characterize positivity and vanishing of the Lyapunov exponent for this model for arbitrary distributions of the random phases. This includes Bernoulli distributions, where in certain cases a finite number of critical spectral values, with vanishing Lyapunov exponent, exists. Thus, we prove that for all non-trivial distributions the model has no absolutely continuous spectrum. For non-singular distributions of the random phases, we show strong spectral localization, i.e. the spectrum is pure point almost surely with exponentially decaying eigenfunctions. Moreover, if the random phases have an absolutely continuous distribution with bounded density, the model is shown to be dynamically localized, i.e. the probability of finding an electron in a high energy state is exponentially small for all time
Recommended Citation
Hamza, Eman F., "Localization Properties For The Unitary Anderson Model" (2007). All ETDs from UAB. 3710.
https://digitalcommons.library.uab.edu/etd-collection/3710