Consider a Schrodinger operator $H=H_0+V$, where $H_0$ is a $L_2$ realization of the negative Laplace operator $-\Delta$ with homogeneous Neumann boundary condition on a bounded, smooth domain $\Omega\subset\mathbb{R}^n$ and $V$ is a smooth real-valued function on $\overline{\Omega}$. I am interested in the following
Problem
Find nontrivial necessary and/or sufficient conditions for $V$ under which $H$ is nonnegative i.e.
\begin{align} \int_{\Omega}|\nabla u|^2+Vu^2\geq0 \end{align} for every smooth function $u$ satisfying the Neumann boundary condition.
It is known that the operator $H$ is self-adjoint and has a compact resolvent, hence by the appropriate version of the spectral theorem the spectrum of $H$ is real and discrete and the above inequality is equivalent to the claim that the smallest eigenvalue of $H$ is nonnegative.
A trivial sufficient condition is that $V$ is nonnegative. A trivial necessary condition is that $\int_{\Omega}V\geq 0$ or more generally that \begin{align*} \int_{\Omega}(V+\lambda)\phi^2\geq0 \end{align*} for any $\lambda$ - eigenvalue of $H_0$ and $\phi$ - corresponding eigenfunction.
What happens when there exists $x_0\in\Omega$ such that $V(x_0)<0$? Can $H$ be still nonnegative? I will appreciate any comments/references even for the one-dimensional case $\Omega=(0,1)$.
