Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Mark S. Ashbaugh and Rafael Benguria
Journal: Proc. Amer. Math. Soc. 105 (1989), 419-424
MSC: Primary 81C05; Secondary 34B25
MathSciNet review: 942630
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 81C05, 34B25

Retrieve articles in all journals with MSC: 81C05, 34B25


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0942630-X
PII: S 0002-9939(1989)0942630-X
Keywords: Schrödinger operators, eigenvalue gaps
Article copyright: © Copyright 1989 American Mathematical Society