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Consider a Schrödinger operator $$H=-\Delta+V$$ on a nice bounded domain $\Omega\subset\mathbb R^d$ (say, a ball or a cube), and assume for simplicity that $V$ is smooth.

Let $\lambda_D,\lambda_N$ and $\lambda_P$ denote the ground state eigenvalue of $H$ with Dirichlet, Neumann and periodic boundary conditions respectively. By using the variational characterization of the eigenvalues, it is sometimes possible to get trivial one-sided inequalities such as $$\lambda_P\leq\lambda_D.$$ I'm interested in more detailed comparisons between these eigenvalues:

Question. Does there exist results giving bounds on the distances $$|\lambda_P-\lambda_D|,~|\lambda_N-\lambda_D|,~|\lambda_N-\lambda_P|,$$ probably depending on $\Omega$ and $V$?

My interest in this question stems from the following kind of problem: Suppose we have an asymptotic result on the magnitude of $\lambda_D$ on the open ball $B(0,r)$ as $r\to\infty$. Is it sometimes possible to infer from this that $\lambda_N$ and $\lambda_P$ have the same asymptotic behavior?

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    $\begingroup$ this physics heuristics may be of some help: the sensitivity of an eigenvalue (energy level) to a change in boundary condition is called the "Thouless energy", it is given by the inverse of the time needed to propagate across the volume; Thouless compared this eigenvalue change $\Delta$ to the spacing $\delta$ between subsequent eigenvalues; if $\Delta<\delta$ the eigenfunctions are localized, if $\Delta>\delta$ the eigenfunctions are extended. $\endgroup$ Jul 12, 2018 at 18:36
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    $\begingroup$ I think it might be worth an attempt to try to formulate a more specific question along these lines. In this generality, the min-max principle pretty much shows you what is going on (by the way, it of course also implies that $\lambda_N\le\lambda_D$ always). If you have negative potential sitting very close to the boundary, then $\lambda_N$ can take advantage of that while $\lambda_D$ can't ... $\endgroup$ Jul 12, 2018 at 19:27
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    $\begingroup$ (cont'd) For example, in $d=1$, you can get a difference $\lambda_D-\lambda_N \simeq \|V\|_1^2$ in this way, for arbitrarily large $\Omega$. $\endgroup$ Jul 12, 2018 at 19:28

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