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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 $V$?

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    $\begingroup$ If we introduce $v(|x|)=1-V(|x|)$ then what you have is precisely Schrodinger eigenvalue equation (with Neumann BV) .see of instance here $\endgroup$
    – BigM
    Commented Dec 7, 2015 at 6:16
  • $\begingroup$ thank you for the reference. It appears the reference only considers the Dirichlet problem. Any idea where I can find a Neumann version of this? thanks $\endgroup$
    – Math604
    Commented Dec 7, 2015 at 6:30
  • $\begingroup$ This is a separable problem, is it not? $\endgroup$
    – username
    Commented Jun 1, 2021 at 13:25
  • $\begingroup$ apparently i asked this 5 or 6 years ago... i have no clue what i asked or why or .... $\endgroup$
    – Math604
    Commented Jun 2, 2021 at 3:04

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