Jonathan Breuer: Mini Course: An Overview of the Spectral Theory of Schrödinger Type Operators with a Random Decaying Potential

Abstract:

Random Schrödinger operators are models for quantum Hamiltonians of particles in disordered media. As such, their spectral theory has been a topic of intense interest in mathematical physics for several decades. The case of one dimensional operators with a random decaying potential has been introduced in the early 1980's as one where a transition from localized to extended states (still an open problem for extensive randomness in high dimensions) can be rigorously proven. Since then, this model and various related ones have been found to exhibit rich spectral properties and surprising connections to random matrix theory.

In this short series of talks we plan to give an overview of the spectral theory of Schrödinger type operators with a random decaying potential. We shall present the various models, describe connections to random matrix theory and orthogonal polynomials, and review the main results and tools concerning both the infinite and asymptotic spectral theory. The talks are intended for a general mathematical audience at a graduate level. Specifically, we assume previous knowledge of measure theory, probability theory, and functional analysis (including spectral theory).

Schedule:

Lecture 1: Wednesday February 19, 13:15 — 14:15
Lecture 2: Monday February 24, 15:15 — 16: 15
Lecture 3: Wednesday February 26, 13:15 — 14:15