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 constant potentials, 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.