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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Lower bounds of the gap between the first and second eigenvalues of the Schrödinger operator
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by Qi Huang Yu and Jia Qing Zhong PDF
Trans. Amer. Math. Soc. 294 (1986), 341-349 Request permission


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 \[ - \Delta f + Vf = \lambda f,\qquad f{|_{\partial \Omega }} \equiv 0.\] Then ${\lambda _2} - {\lambda _1} \geqslant {\pi ^2}/{d^2}$, where $d$ is the diameter of $\Omega$.
  • 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, DOI 10.1016/0022-1236(76)90004-5
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  • 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
  • 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
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 294 (1986), 341-349
  • MSC: Primary 35P05; Secondary 35J10
  • DOI:
  • MathSciNet review: 819952