Ground states for nonlocal Schrödinger type operators on locally compact abelian groups

Abstract

We find classes of nonlocal operators of Schrödinger type on a locally compact noncompact Abelian group $\mathfrak{G}$, 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 $\mathfrak{G}={\mathbb{Q}}_p^n$ where ${\mathbb{Q}}_p$ is the field of $p$-adic numbers.