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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
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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.

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PII: S 0002-9939(1989)0942630-X
Keywords: Schrödinger operators, eigenvalue gaps
Article copyright: © Copyright 1989 American Mathematical Society