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

• Published in: Proceedings of the American Mathematical Society 1989, Vol.105 (2), p.419-424
• Main Authors: Ashbaugh, Mark S., Benguria, Rafael
• Format: Article
• Language: English
• Subjects: Boundary conditions; Eigenvalues; Exact sciences and technology; Function theory, analysis; Grants; Mathematical methods in physics; Physics; Research article
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-1989-0942630-X

We prove the optimal lower bound λ2−λ1≥3π2/d2{\lambda _2} - {\lambda _1} \geq 3{\pi ^2}/{d^2} for the difference of the first two eigenvalues of a one-dimensional Schrödinger operator −d2/dx2+V(x)- {d^2}/d{x^2} + V(x) with a symmetric single-well potential on an interval of length dd and with Dirichlet boundary conditions. Equality holds if and only if the potential is constant. More generally, we prove the inequality λ2[V1]−λ1[V1]≥λ2[V0]−λ1[V0]{\lambda _2}[{V_1}] - {\lambda _1}[{V_1}] \geq {\lambda _2}[{V_0}] - {\lambda _1}[{V_0}] in the case where V1{V_1} and V0{V_0} are symmetric and V1−V0{V_1} - {V_0} is a single-well potential.