PhD Thesis Defense: "Classical and Quantum Markov Chains Derived from Billiard-like Systems"

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:

1. 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.

2. 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.

Host: Renato Feres