Schrödinger operators on Lie groups with purely discrete spectrum

Abstract

On a Lie group $G$, we investigate the discreteness of the spectrum of Schrödinger 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 $G$.

32 pages

References in corpus (2)

• Schrodinger Operators with Purely Discrete Spectrum
• Local and nonlocal Poincaré inequalities on Lie groups

Cited by in corpus (1)

• Shubin calculi for actions of graded Lie groups