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 $V$?

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 $ \mathbb R^N$ and consider the eigenvalue problem

\begin{cases} - \Delta u(x) + u(x) - V(|x|) u(x) = \lambda u(x) & \mbox{in } B \\\partial_\nu u=0 & \partial B. \end{cases}

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 $V$?