Schr\"odinger operators on Lie groups with purely discrete spectrum

TL;DR

This paper characterizes when Schr"odinger operators on Lie groups have purely discrete spectra, providing necessary and sufficient conditions, especially for polynomial and Muckenhoupt class potentials, with implications for weighted sub-Laplacians.

Contribution

It offers the first comprehensive criteria for the discreteness of spectra of Schr"odinger operators on Lie groups, including explicit conditions for specific potential classes.

Findings

Necessary and sufficient conditions for discrete spectrum

Explicit characterizations for polynomial potentials

Results applicable to weighted sub-Laplacians

Abstract

On a Lie group $G$, we investigate the discreteness of the spectrum of Schr\"odinger operators of the form $\mathcal{L} +V$, where $\mathcal{L}$ is a subelliptic sub-Laplacian on $G$ and the potential $V$ is a locally integrable function which is bounded from below. We prove general necessary and sufficient conditions for arbitrary potentials, and we obtain explicit characterizations when $V$ is a polynomial on $G$ or belongs to a local Muckenhoupt class. We finally discuss how to transfer our results to weighted sub-Laplacians on $G$.