Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials

Abstract:

We prove the optimal lower bound ${\lambda _2} - {\lambda _1} \geq 3{\pi ^2}/{d^2}$ for the difference of the first two eigenvalues of a one-dimensional Schrödinger operator $- {d^2}/d{x^2} + V(x)$ with a symmetric single-well potential on an interval of length $d$ and with Dirichlet boundary conditions. Equality holds if and only if the potential is constant. More generally, we prove the inequality ${\lambda _2}[{V_1}] - {\lambda _1}[{V_1}] \geq {\lambda _2}[{V_0}] - {\lambda _1}[{V_0}]$ in the case where ${V_1}$ and ${V_0}$ are symmetric and ${V_1} - {V_0}$ is a single-well potential.