  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.
