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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: 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] H. Brascamp and E. Liep, On extensions of the Brunn-Minkowski and Prëkopa-Leindler theorems, including inequalities for log concave function, and with an application to diffusion equation, J. Funct. Anal. 22 (1976), 366-389. MR 0450480 (56:8774)
  • [2] R. Courant and D. Hilbert, Methods of mathematical physics, vol. I, Interscience, New York, 1953. MR 0065391 (16:426a)
  • [3] Bun Wong, Shing-Tung Yau and Stephen S.-T. Yau, An estimate of the first two eigenvalues in the Schrödinger operator (to appear). 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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Article copyright: © Copyright 1986 American Mathematical Society

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