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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Lower bounds of the gap between the first and second eigenvalues of the Schrödinger operator

Authors: Qi Huang Yu and Jia Qing Zhong
Journal: Trans. Amer. Math. Soc. 294 (1986), 341-349
MSC: Primary 35P05; Secondary 35J10
MathSciNet review: 819952
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper the authors prove the following theorem:

Let $ \Omega $ be a smooth strictly convex bounded domain in $ {R^n}$ and $ V:\Omega \to R$ a nonnegative convex function. Suppose $ {\lambda _1}$ and $ {\lambda _2}$ are the first and second nonzero eigenvalues of the equation

$\displaystyle - \Delta f + Vf = \lambda f,\qquad f{\vert _{\partial \Omega }} \equiv 0.$

Then $ {\lambda _2} - {\lambda _1} \geqslant {\pi ^2}/{d^2}$, where $ d$ is the diameter of $ \Omega $.

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

  • [1] Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366–389. MR 0450480 (56 #8774)
  • [2] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391 (16,426a)
  • [3] Alexander Grigor′yan and Shing Tung Yau (eds.), Surveys in differential geometry. Vol. IX, Surveys in Differential Geometry, vol. 9, International Press, Somerville, MA, 2004. Eigenvalues of Laplacians and other geometric operators. MR 2184990 (2006g:58001)
  • [4] Jia-Qing Zong and Hong-Zhang Yang, Estimates of the first eigenvalue of Laplace operator on compact Riemannian manifolds, Sci. Sinica Ser. A 9 (1983), 812-820. (Chinese)

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Additional Information

PII: S 0002-9947(1986)0819952-8
Article copyright: © Copyright 1986 American Mathematical Society

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