Assume $\Omega$ is a non-compact region or manifold with dimension $\geq4$. Let $H=-\Delta+V$ be Schrodinger operator. Here $V$ is a (smooth)function.
I know that if $V\geq c>0$ or $V\to c>0$, then $0$ does not locate in the essential spectrum of $H$.s
Q : Is there any work to consider the negative case, i.e. $V>-c$, here $c>0$ is a constant, with what condition on $V$, we also have that $0$ locates in the discrete spectrum of $H$