Ground States for Nonlocal Schr\"odinger Type Operators on Locally Compact Abelian Groups

TL;DR

This paper investigates the existence of ground states for nonlocal Schr"odinger-type operators on locally compact Abelian groups, providing explicit conditions especially for p-adic number fields, with implications for stochastic processes.

Contribution

It introduces new classes of nonlocal operators on Abelian groups and establishes conditions for ground state existence, including for p-adic groups, extending previous results.

Findings

Existence of ground states for certain nonlocal Schr"odinger operators on Abelian groups.

Explicit criteria for ground state existence on p-adic groups.

Connection between recurrent random walks and ground state existence.

Abstract

We find classes of nonlocal operators of Schr\"odinger type on a locally compact noncompact Abelian group $G$, for which there exists a ground state. In particular, such a result is obtained for the case where the principal part of our operator generates a recurrent random walk. Explicit conditions for the existence of a ground state are obtained for the case $G =\mathbb Q_p^n$ where $\mathbb Q_p$ is the field of $p$-adic numbers.