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 := -d/dx (p(x)du/dx) + V(x)u = lambda u, x is an element of [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....

Automated summary

This passage characterizes the potential-energy functions V(x) that minimize the gap Gamma between the two lowest Sturm-Liouville eigenvalues. The study reveals that under the assumptions of convexity or single-well form, constant potentials can uniquely minimize Gamma for convex V, and a sharp bound of Gamma > 2.04575... is obtained for single-well potentials without restrictions on the position of the minimum.