Harmonic Analysis and Random Schr\"odinger Operators

TL;DR

This survey explores the interplay between harmonic analysis and properties of Schr"odinger operators, focusing on unique continuation and spectral properties, bridging classical and modern mathematical questions.

Contribution

It provides a comprehensive overview of harmonic analysis techniques applied to Schr"odinger operators, highlighting recent developments and open problems in the spectral theory context.

Findings

Analysis of unique continuation properties for Schr"odinger operators

Characterization of eigenvalue and eigenfunction behaviors

Connections between harmonic analysis and spectral theory

Abstract

This survey is based on a series of lectures given during the \emph{School on Random Schr\"odinger Operators} and the \emph{International Conference on Spectral Theory and Mathematical Physics} at the Pontificia Universidad Catolica de Chile, held in Santiago in November 2014. As the title suggests, the presented material has two foci: Harmonic analysis, more precisely, unique continuation properties of several natural function classes and Schr\"odinger operators, more precisely properties of their eigenvalues, eigenfunctions and solutions of associated differential equations. It mixes topics from (rather) pure to (rather) applied mathematics, as well as classical questions and results dating back a whole century to very recent and even unpublished ones. The selection of material covered is based on the selection made for the minicourse, and is certainly a personal choice corresponding to the research interests of the authors.