I am sure this is extremely well known but I have been digging a bit and I can't find what I need.  Consider $B$ to be the unit ball in $ R^N$ and consider the eigenvalue problem
$$- \Delta u(x) + u(x) - V(|x|) u(x) = \lambda u(x) \quad  \mbox{ in } \quad B $$  with $ \partial_\nu u=0$ on $ \partial B$.  Here $V(r)$ is some fixed positive function.  Let $ \lambda_1$ denote the first eigenvalue of this problem and let $ \lambda_2$ denote the second.   Are there available lower estimates on $ \lambda_2 - \lambda_1$ in terms of some quantities involving $V$?