Consider $ \Omega$ a smooth bounded domain in $ \mathbb R^N$.  

I am interested in the gap between the first and second eigenvalues of the operator $ -\Delta + V(x)$. Let $ \phi\_1>0$ and $ \phi\_2$ be the first and second eigenfunction for this operator and so
$$ -\Delta \phi\_i + V(x) \phi\_i = \mu\_i \phi\_i $$ in $ \Omega$ with $ \phi\_i=0$ on $ \partial \Omega$.


So I am interested in getting a lower bound on $ \mu\_2 - \mu\_1$.


The ‘fundamental gap conjecture’ is related to an explicit lower bound on this quantity. My interest is to not assume $V$ convex (many consider $V$ convex) but I can assume ‘semi-convex’; ie. $V(x)+ c \lvert x\rvert^2$ convex for some $C>0$.


My interest is any sort of explicit positive lower bound on $\mu\_2-\mu\_1$; but I don't care at all if its optimal.



Fundamental gap for Schrödinger operator
You want to consult [Proof of the Fundamental Gap Conjecture](https://arxiv.org/abs/1006.1686) where it is shown that
$$
\mu\_2 - \mu\_1 \geq \frac{3\pi^2}{D^2}
$$
where $D$ is the diameter of $\Omega $.


