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 λ 2 − λ 1 ≥ 3 π 2 / d 2 {\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 ) - {d^2}/d{x^2} + V(x) with a symmetric single-well potential on an interval of length d d and with Dirichlet boundary conditions. Equality holds if and only if the potential is constant. More generally, we prove the inequality λ 2 [ V 1 ] − λ 1 [ V 1 ] ≥ λ 2 [ V 0 ] − λ 1 [ V 0 ] {\lambda _2}[{V_1}] - {\lambda _1}[{V_1}] \geq {\lambda _2}[{V_0}] - {\lambda _1}[{V_0}] in the case where V 1 {V_1} and V 0 {V_0} are symmetric and V 1 − V 0 {V_1} - {V_0} is a single-well potential.