Advisory Committee Chair
Gunter Stolz
Advisory Committee Members
Timothy Ferguson
Ryoichi Kawai
Boris Kunin
Rudi Weikard
Document Type
Dissertation
Date of Award
2015
Degree Name by School
Doctor of Philosophy (PhD) College of Arts and Sciences
Abstract
In 1958, P. W. Anderson introduced a model to explain the quantum mechanical effects of disorder. In subsequent years, much research has been done on the model which carries his name, the Anderson model. The study of this model led to the discovery of the phenomenon of Anderson localization, i.e the suppression of electron transport due to the system's disorder. We show that, starting with a model for a general system of quantum harmonic oscillators, the Anderson model results from a particular method of introducing disorder into the system. By exploring additional options for introducing disorder into the system, we can discover new models worth studying. Namely, these models are called the Random Mass Laplacian and the Random Edge Laplacian. After a brief survey of the basic properties of these models, we show that the one-dimensional models exhibit spectral localization, i.e almost sure pure point spectrum, via the use of transfer matrices and the Lyapunov exponent. Additionally, we demonstrate that, for both models, the Lypunov exponent goes to zero linearly as the energy goes to zero. Further, we begin the process of showing localization for these models in higher dimensions via the well-known Fractional Moment Method. We establish an a priori bound on the fractional moments of each model's Green function. In order to proceed further, we introduce a relationship between these two models for the graph consisting of $\Z^d$ and next-neighbor edges. Using these results, we prove that, for certain fractional moments, the fractional moments of these Green functions decay exponentially in the distance between vertices. For the Random Mass Laplacian, this decay is shown on general graphs. However, at this time, this result is limited to $\Z^d$ in the case of the Random Edge Laplacian. This exponential decay has become a well-accepted criterion for localization.
Recommended Citation
Besing, Kyle, "Spectral Properties and Localization of Two Random Laplacians on Graphs" (2015). All ETDs from UAB. 1154.
https://digitalcommons.library.uab.edu/etd-collection/1154