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.

Committee Chair

Renato Feres

Committee Members

Yanli Song

Comments

Update embargo

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

Spring 5-15-2022

Language

English (en)

Author's ORCID

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

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Mathematics Commons

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