All ETDs from UAB

Advisory Committee Chair

Gunter Stolz

Advisory Committee Members

Richard Brown

Ioulia Karpechina

Ryoichi Kawai

Boris Kunin

Document Type


Date of Award


Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences


The recent study of Schr¨odinger operators with random potentials has provided significant insight into our understanding of the electronic properties of disordered media. This work presents new results concerning the spectrum of a class of random operators called displacement models. Such models may be used to model solids in which the positions of the individual atoms are randomly perturbed from an ideal periodic lattice. In particular, we will provide a characterization of the almost sure spectral minimum of the random displacement model, bounds on the integrated density of states, and a rigourous proof of the lack of classical Lifshitz tails under suitable assumptions on the random parameters. The fundamental tool used throughout the work is a quite general phenomenon in the spectral theory of Neumann problems, which we dub “bubbles tend to the boundary.” How should a given compactly supported potential be placed into a bounded domain so as to minimize or maximize the first Neumann eigenvalue of the Schrdinger operator on this domain? For square or rectangular domains and reflection symmetric potentials, we show that the first Neumann eigenvalue is minimized when the potential sits in one of the corners of the domain and is maximized when it sits in the center of the domain. With different methods we also show a corresponding result for smooth strictly convex domains.



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