The fundamental gap conjecture for convex Schrödinger operators

Let $\Omega\subset\mathbb{R}^n$ be a bounded convex domain of diameter $D$, and let $V$ be a weakly convex potential. The Dirichlet eigenvalues $\lambda_1\leq\lambda_2\leq\cdots$ of the Schrödinger operator $-\Delta+V$ then satisfy

$$\lambda_2-\lambda_1\geq \frac{3\pi^2}{D^2}.$$

Fundamental gap conjecture. The eigenvalues satisfy the inequality above.

The bound asserts that the spectral gap is at least the gap for an interval of length $D$ with zero potential. The paper proves this conjecture, so it is included as solved.