Spectral Gaps from Ordered to Disordered Systems

Citation

Lindner, John Florian (1989) Spectral Gaps from Ordered to Disordered Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/gv0b-re42. https://resolver.caltech.edu/CaltechETD:etd-08172005-161212

Abstract

As is well known, the allowed energies of periodic electronic systems and the allowed frequencies of periodic elastic systems form banded sets (at least for certain idealized models). Recent work, by Werner Kirsch and others, demonstrates that this band-gap structure persists in disordered versions of these periodic systems. Here, I extend this result by showing that for specific "point" interactions, the spectrum of a generic disordered system is the union of the spectra of all possible pure systems formed from it. This permits the explicit construction of these spectral sets.

This result is the outgrowth of a perspective I call "growing disorder." The idea is to evolve, or "grow," an ordered array (whose spectrum is known) into a disordered array (whose spectrum is sought). The trick is to evolve the spectrum along with it. The approach is very visual, lends itself readily to graphical presentation, and accounts in part for the unconventional but appropriate look of this thesis.

The unconventional style also reflects an attempt to make the material easily accessible to a physics audience. It is inspired by the way in which physicists informally communicate ideas, namely, with words and picture in front of a blackboard. Each page, or set of text and graphics is a unit to be assimilated before proceeding onto the next unit. There is thus no unique path through the thesis.

An intuitive and straightforward approach, constructive proofs, an informal style, and some ingenuity simply communicate the ideas herein. However, the condensation inherent in the graphical presentation demands significant reader engagement!

Thesis Files