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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.

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    $\begingroup$ does "semiconvex" exclude a double-well potential? if it does not, there is nothing from preventing two nearly degenerate eigenvalues, one from each well. $\endgroup$ Apr 9, 2017 at 13:43
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    $\begingroup$ not sure what nearly degenerate eigenvalues means? i assume it means you can make $ \mu_2 = \mu_1$ arbitrarily small? (in any case, no i can't exclude the double well). $\endgroup$
    – Math604
    Apr 9, 2017 at 16:48
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    $\begingroup$ in a symmetric double well potential the difference $|\mu_2-\mu_1|$ is exponentially small in the thickness of the barrier that separates the two wells. $\endgroup$ Apr 9, 2017 at 18:08
  • $\begingroup$ I meant to write $ \mu_2-\mu_1 $ small... okay, thanks for the result. This is what i was looking for. $\endgroup$
    – Math604
    Apr 9, 2017 at 21:21

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You want to consult Proof of the Fundamental Gap Conjecture where it is shown that $$ \mu_2 - \mu_1 \geq \frac{3\pi^2}{D^2} $$ where $D$ is the diameter of $\Omega $.

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  • $\begingroup$ doesn't this assume convex $\Omega$? $\endgroup$ Apr 10, 2017 at 6:29
  • $\begingroup$ Yes, sorry it does. $\endgroup$
    – Paul Bryan
    Apr 10, 2017 at 7:00
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    $\begingroup$ Without the convexity assumption, is such a bound on the gap possible? $\endgroup$
    – Paul Bryan
    Apr 12, 2017 at 1:41
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    $\begingroup$ I've been told that a counter example to a lower bound in the case no convexity of $\Omega$ is assumed is given by two balls joined by a thin neck. Perhaps this is similar to the double well potential? But note here the problem is with the domain and not the potential. $\endgroup$
    – Paul Bryan
    Apr 14, 2017 at 8:49

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