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.