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

  • Mark S. Ashbaugh
  • Rafael Benguria
Conference paper
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.

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Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Mark S. Ashbaugh
    • 1
  • Rafael Benguria
    • 2
  1. 1.Department of MathematicsUniversity of MissouriColumbia
  2. 2.Departamento de Física Facultad de Ciencias Físicas y MatemáticasUniversidad de ChileSantiagoChile

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