Fundamental gap conjecture

Let $\Omega \subset \mathbb R^n$ be a bounded convex domain of diameter $D$ and $V:\Omega \to \mathbb{R}$ a convex potential. The eigenvalues of the Schrödinger operator $-\Delta + V$ satisfy

$$\lambda_{2}(\Omega)-\lambda_{1}(\Omega)\geq \frac{3\pi^{2}}{D^{2}}.$$

Fundamental gap conjecture. The fundamental gap is at least $3\pi^2/D^2$.

The conjecture gives a sharp lower bound for the difference between the first two Dirichlet eigenvalues on bounded convex Euclidean domains with convex potential. It was proved in 2011 by Andrews and Clutterbuck using a two-point maximum principle.