Skip to main content
SpringerLink
Log in

Optimal bounds for ratios of eigenvalues of one-dimensional Schrödinger operators with Dirichlet boundary conditions and positive potentials

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Consider the Schrödinger equation −u″+V(x)u=λu on the intervalI⊂ℝ, whereV(x)≧0 forx∈I and where Dirichlet boundary conditions are imposed at the endpoints ofI. We prove the optimal bound

$$\frac{{\lambda _n }}{{\lambda _1 }} \leqq n^2 for n = 2,3,4,...$$

on the ratio of then th eigenvalue to the first eigenvalue for this problem. This leads to a complete treatment of bounds on ratios of eigenvalues for such problems. Extensions of these results to singular problems are also presented. A modified Prüfer transformation and comparison techniques are the key elements of the proof.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ashbaugh, M. S., Benguria, R. D.: Best constant for the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with positive potentials. Proc. Am. Math. Soc.99, 598–599 (1987)

    Google Scholar 

  2. Ashbaugh, M. S., Benguria, R. D.: On the ratio of the first two eigenvalues of Schrödinger operators with positive potentials, pp. 16–25. In: Differential equations and mathematical physics. Knowles, I. W., Saitō, Y. (eds.), Lecture Notes in Mathematics, vol.1285. Berlin, Heidelberg, New York: Springer 1987

    Google Scholar 

  3. Ashbaugh, M. S., Benguria, R. D.: Optimal lower bounds for eigenvalue gaps for Schrödinger operators with symmetric single-well potentials and related results, pp. 134–145. In: Maximum principles and eigenvalue problems in partial differential equations. Schaefer, P. W. (ed.), Pitman Research Notes in Mathematics Series, vol.175, Harlow, Essex, UK: Longman Scientific and Technical 1988

    Google Scholar 

  4. Barnsley, M. F.: Lower bounds for quantum-mechanical energy levels, J. Phys.A 11, 55–68 (1978)

    Google Scholar 

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

    Google Scholar 

  6. Birkhoff, G., Rota, G.-C.: Ordinary differential equations, 3rd ed. New York: John Wiley 1978

    Google Scholar 

  7. Chiti, G.: A bound for the ratio of the first two eigenvalues of a membrane, SIAM J. Math Anal.14, 1163–1167 (1983)

    Google Scholar 

  8. Combes, J. M., Duclos, P., Seiler, R.: Krein's formula and one-dimensional multiple-well. J. Funct. Anal.52, 257–301 (1983)

    Google Scholar 

  9. Courant, R., Hilbert, D.: Methods of mathematical physics, Vol.I, New York: Interscience Publishers 1953

    Google Scholar 

  10. Crandall, R. E., Reno, M.H.: Ground state energy bounds for potentials |x|v. J. Math. Phys.23, 64–70 (1982)

    Google Scholar 

  11. Crum, M. M.: Associated Sturm-Liouville systems. Quart. J. Math. Oxford6, 121–127 (1955)

    Google Scholar 

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

    Google Scholar 

  13. Harrell, E. M.: unpublished (1982)

  14. Kuttler, J. R., Sigillito, V. G.: Eigenvalues of the Laplacian in two dimensions. SIAM Rev.26, 163–193 (1984)

    Google Scholar 

  15. Mahar, T. J., Willner, B. E.: An extremal eigenvalue problem. Commun. Pure App. Math.29, 517–529 (1976)

    Google Scholar 

  16. Payne, L. E., Pólya, G., Weinberger, H. F.: Sur le quotient de deux frequences propres consecutives. C. R. Acad. Sci. Par.241, 917–919 (1955)

    Google Scholar 

  17. Payne, L. E., Pólya, G., Weinberger, H. F.: On the ratio of consecutive eigenvalues. J. Math. and Phys.35, 289–298 (1956)

    Google Scholar 

  18. Protter, M. H.: Can one hear the shape of a drum? revisited. SIAM Rev.29, 185–197 (1987)

    Google Scholar 

  19. Protter, M. H.: Lower bounds for the spectrum of second order operators and systems of operators, pp. 82–93, and Universal inequalities for eigenvalues, pp. 111–120, In: Maximum principles and eigenvalue problems in partial differential equations. Schaefer, P. W., (ed.), Pitman Research Notes in Mathematics Series, vol.175, Harlow, Essex, United Kingdom: Longman Scientific and Technical 1988

    Google Scholar 

  20. Reed, M., Simon, B.: Methods of modern mathematical physics, Vol IV: Analysis of operators. New York: Academic Press 1978

    Google Scholar 

  21. Singer, I. M., Wong, B, Yau, S.-T., Yau, S. S.-T: An estimate of the gap of the first two eigenvalues in the Schrödinger operator. Ann. Scuola Norm. Sup. Pisa12, 319–333 (1985)

    Google Scholar 

  22. Thompson, C. J.: On the ratio of consecutive eigenvalues in N-dimensions. Stud. Appl. Math.48, 281–283 (1969)

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Mark S. Ashbaugh

  2. Departmento de Física, F.C.F.M., Universidad de Chile, Casilla, 487-3, Santiago, Chile

    Rafael D. Benguria

Authors
  1. Mark S. Ashbaugh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rafael D. Benguria
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by B. Simon

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ashbaugh, M.S., Benguria, R.D. Optimal bounds for ratios of eigenvalues of one-dimensional Schrödinger operators with Dirichlet boundary conditions and positive potentials. Commun.Math. Phys. 124, 403–415 (1989). https://doi.org/10.1007/BF01219657

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01219657

Keywords

Search

Navigation