ORCID

http://orcid.org/0000-0002-6098-3585

Date of Award

Spring 5-15-2022

Author's School

Graduate School of Arts and Sciences

Author's Department

Mathematics

Degree Name

Doctor of Philosophy (PhD)

Degree Type

Dissertation

Abstract

Random billiards are a class of random dynamical systems related to dynamical billiards. We extend the study of random billiards and their associated Markov chains in two new directions. First, we introduce a new class of billiard-like systems called lensed billiards, which introduce a step potential to the usual billiard set-up, and conduct an exploratory study of random lensed billiards where we are mainly interested in how the newly-introduced potential parameter relates to the spectral gap and set of moments of the Markov operator associated to the random lensed system.

Second, we recast the mathematical set-up of random billiards to the operator theoretic framework used in open quantum systems, which we use to obtain a description of the quantum counterparts of the Markov chains associated to the random billiards we are interested in. When viewed from this perspective, we see that scattering theory plays a prominent role in the quantum case.

Language

English (en)

Chair and Committee

Renato Feres

Committee Members

Yanli Song

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