Year of Publication

2021

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department/School/Program

Mathematics

First Advisor

Dr. Peter Hislop

Abstract

We provide three proofs on different, but related models in the field of random Schrödinger operators. All three results are motivated by the desire to extend results and techniques on eigenvalue statistics or Minami estimates (an essential ingredient Poisson eigenvalue statistics).

Chapters 2 and 4 are explorations of the only two known techniques for proving Minami estimates for continuum Minami estimates. In Chapter 2, we provide an alternative and simplified proof of Klopp that holds in d = 1. Chapter 4 is an application of the techniques of Dietlein and Elgart to prove a Minami estimate for finite rank lattice models, which is an improvement on known results. Chapter 3 is an improvement on a result of Dolai and Krishna, in which we show the statistics for a RSO with a decaying potential is the same as the free Laplacian for decay down to |n|-α for α>1.

The first chapter is a brief, general introduction to the Anderson model and relevant concepts in the field random Schrödinger operators. Each subsequent chapter has a more specific introduction before proceeding with the results.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2021.409

Funding Information

University of Kentucky, Mathematics Department Teaching Assistantship, 2014-2020

University of Kentucky, Mathematics Department Summer Research Fellowship, 2017

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