Skip to main content
SpringerLink
Log in

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

  • Conference paper
  • First Online:
Differential Equations and Mathematical Physics

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

Abstract

We survey current knowledge on the ratio, λ2/λ1, 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 λ2/λ1≤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)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ashbaugh, M.S., and R. Benguria, Best Constant for the Ratio of the First Two Eigenvalues of One-Dimensional Schrödinger Operators with Positive Potentials, to appear in Proc. Amer. Math. Soc.

    Google Scholar 

  2. Benguria, R., A note on the gap between the first two eigenvalues for the Schrödinger operator, J. Phys. A: Math. Gen. 19 (1986) 477–478.

    Article  MathSciNet  MATH  Google Scholar 

  3. Brands, J.J.A.M., Bounds for the Ratios of the First Three Membrane Eigenvalues, Arch. Rat. Mech. Anal. 16 (1964) 265–268.

    Article  MathSciNet  MATH  Google Scholar 

  4. Crum, M.M., Associated Sturm-Liouville systems, Quart. J. Math. Oxford (2) 6 (1955) 121–127.

    Article  MathSciNet  MATH  Google Scholar 

  5. Deift, P.A., Applications of a commutation formula, Duke J. Math. 45 (1978) 267–310.

    Article  MathSciNet  MATH  Google Scholar 

  6. DeVries, H.L., On the Upper Bound for the Ratio of the First Two Membrane Eigenvalues, Zeitschrift für Naturforschung 22A (1967) 152–153.

    MathSciNet  Google Scholar 

  7. Harrell, E.M., unpublished (1982).

    Google Scholar 

  8. Hile, G.H., and M.H. Protter, Inequalities for Eigenvalues of the Laplacian, Indiana Math. J. 29 (1980) 523–538.

    Article  MathSciNet  MATH  Google Scholar 

  9. Marchenko, V.A., The construction of the potential energy from the phases of the scattered waves, Dokl. Akad. Nauk. SSSR 104 (1955) 695–698 [Math. Rev. 17 (1956) 740].

    MathSciNet  Google Scholar 

  10. Payne, L.E., G. Pólya, and H.F. Weinberger, Sur le quotient de deux fréquences propres consécutives, Comptes Rendus Acad. Sci. Paris 241 (1955) 917–919.

    MATH  Google Scholar 

  11. Payne, L.E., G. Pólya, and H.F. Weinberger, On the Ratio of Consecutive Eigenvalues, J. Math. and Phys. 35 (1956) 289–298.

    Article  MathSciNet  MATH  Google Scholar 

  12. Singer, I.M., Wong, B., S.-T. Yau, and S.S.-T. Yau, An estimate of the gap of the first two eigenvalues in the Schrödinger operator, Ann. Scuola Norm. Sup. Pisa (series 4) 12 (1985) 319–333.

    MathSciNet  MATH  Google Scholar 

  13. Thompson, C.J., On the Ratio of Consecutive Eigenvalues in N-Dimensions, Stud. Appl. Math. 48 (1969) 281–283.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Missouri, 65211, Columbia, MO

    Mark S. Ashbaugh

  2. Departamento de Física Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487/3, Santiago, Chile

    Rafael Benguria

Authors
  1. Mark S. Ashbaugh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rafael Benguria
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Ian W. Knowles Yoshimi Saitō

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag

About this paper

Cite this paper

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

Download citation

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

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18479-9

  • Online ISBN: 978-3-540-47983-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation