# Fundamental gap for Schrödinger operator

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.

• does "semiconvex" exclude a double-well potential? if it does not, there is nothing from preventing two nearly degenerate eigenvalues, one from each well. Apr 9, 2017 at 13:43
• 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). Apr 9, 2017 at 16:48
• 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. Apr 9, 2017 at 18:08
• I meant to write $\mu_2-\mu_1$ small... okay, thanks for the result. This is what i was looking for. Apr 9, 2017 at 21:21

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 $\Omega$.
• doesn't this assume convex $\Omega$? Apr 10, 2017 at 6:29
• 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. Apr 14, 2017 at 8:49