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On the ratio of the first two eigenvalues of Schrödinger operators with positive potentials

Part of the Lecture Notes in Mathematics book series (LNM,volume 1285)

Abstract

We survey current knowledge on the ratio, λ21, of the first two eigenvalues of the Schrödinger operator HV=-Δ+V(x) on the region Ω ⊂ ℝn with Dirichlet boundary conditions and non-negative potentials. We discuss the Payne-Pólya-Weinberger conjecture for H0=−Δ and generalize the conjecture to Schrödinger operators. Lastly, we present our recent result giving the best possible upper bound λ21≤4 for one-dimensional Schrödinger operators with nonnegative potentials and discuss some extensions of this result.

Partially supported by grants from the Research Council of the Graduate School, University of Missouri-Columbia and the Programa de las Naciones Unidas para el Desarrollo (PNUD grant CHI-84-005)

Partially supported by the Departamento de Investigación y Bibliotecas de la Universidad de Chile (Grant E-1959-8522)

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© 1987 Springer-Verlag

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Ashbaugh, M.S., Benguria, R. (1987). On the ratio of the first two eigenvalues of Schrödinger operators with positive potentials. In: Knowles, I.W., Saitō, Y. (eds) Differential Equations and Mathematical Physics. Lecture Notes in Mathematics, vol 1285. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0080577

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  • DOI: https://doi.org/10.1007/BFb0080577

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18479-9

  • Online ISBN: 978-3-540-47983-3

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