Optimal bounds on the fundamental spectral gap with single-well potentials

We characterize the potential-energy functions $V(x)$ that minimize the gap $\Gamma$ between the two lowest Sturm-Liouville eigenvalues for \[ H(p,V) u := -\frac{d}{dx} \left(p(x)\frac{du}{dx}\right)+V(x) u = \lambda u, \quad\quad x\in [0,\pi ], \] where separated self-adjoint boundary conditions are imposed at end points, and $V$ is subject to various assumptions, especially convexity or having a "single-well" form. In the classic case where $p=1$ we recover with different arguments the result of Lavine that $\Gamma$ is uniquely minimized among convex $V$ by the constant, and in the case of single-well potentials, with no restrictions on the position of the minimum, we obtain a new, sharp bound, that $\Gamma > 2.04575\dots$.