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 $H$
Obviously, the potential need to be bigger than some positive constant "most of the time". I did not see this exact question to be addressed in the literature. However, a related question: what is the most general condition on $V$ which guarantees that the spectrum of $H$ is discrete, was intensively studied, see
Kondratʹev, V. A.; Shubin, M. A. Conditions for the discreteness of the spectrum for Schrödinger operators on a manifold. Funct. Anal. Appl. 33 (1999), no. 3, 231–232 (2000)
Roughly, the condition is that there is a "small" subset outside of which the potential grows to infinity as $x\to \infty$.
