Skip to main content
SpringerLink
Log in

Pleijel’s theorem for Schrödinger operators with radial potentials

  • Published:
Annales mathématiques du Québec Aims and scope Submit manuscript

Abstract

In 1956, Pleijel gave his celebrated theorem showing that the inequality in Courant’s theorem on the number of nodal domains is strict for large eigenvalues of the Laplacian. This was a consequence of a stronger result giving an asymptotic upper bound for the number of nodal domains of the eigenfunction as the eigenvalue tends to \(+\infty \). A similar question occurs naturally for the case of the Schrödinger operator. The first significant result has been obtained recently by the first author for the case of the harmonic oscilllator. The purpose of this paper is to consider more general potentials which are radial. We will analyze either the case when the potential tends to \(+\infty \) or the case when the potential tends to zero, the considered eigenfunctions being associated with the eigenvalues below the essential spectrum.

Résumé

En 1956, \(\AA \). Pleijel a démontré qu’il peut seulement y avoir un nombre fini d’égalités dans le théorème de Courant sur les fonctions propres du laplacien. Ce résultat était accompagné d’une borne supérieure sur le nombre de domaines nodaux des fonctions propres lorsque les valeurs propres tendent vers l’infini. Il est naturel de supposer qu’un résultat similaire peut être démontré dans le cas de l’opérateur de Schrödinger. Le premier résultat significatif dans cette direction a été obtenu par le premier auteur pour l’oscillateur harmonique. L’objectif de cet article est de traiter le cas de potentiels radiaux plus généraux. Nous traiterons les cas où le potentiel tend vers \(+\infty \) ou les cas où le potentiel tend vers zéro, les fonctions propres correspondant à la partie discrète du spectre.

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

Notes

  1. See also [10].

  2. This condition appears when applying Weyl’s formula given by Theorem 4.2 from [25]. At least under stronger assumptions on the regularity of v for \(r\rightarrow +\infty \), it should be possible to assume \(m>0\).

References

  1. Beals, R.: A general calculus of pseudo-differential operators. Duke Math. J. 42, 1–42 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bérard, P., Helffer, B.: On the nodal patterns of the 2D isotropic quantum harmonic oscillator. arXiv:1506.02374 (2015)

  3. Bérard, P., Meyer, D.: Inégalités isopérimétriques et applications. Annales scientifiques de l’École Normale Supérieure, Sér. 4 15(3), 513–541 (1982)

    Article  MATH  Google Scholar 

  4. Bourgain, J.: On Pleijel’s nodal domain theorem. Int. Math. Res. Not. 6, 1601–1612 (2015). arXiv:1308.4422

    MathSciNet  MATH  Google Scholar 

  5. Charron, P.: Théorème de Pleijel pour l’oscillateur harmonique quantique. Mémoire de Maîtrise. Université de Montréal (2015)

  6. Charron, P.: A Pleijel-type theorem for the quantum harmonic oscillator. J. Spectr. Theory arXiv:1512.07880 (2016) (to appear)

  7. Courant, R.: Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke. Nachr. Ges. Göttingen 81–84 (1923)

  8. Cowan, C.: Optimal Hardy inequalities for general elliptic operators with improvements. Commun. Pure Appl. Math. 9, 109–140 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Cwikel, M.: Weak type estimates for singular values and the number of bound states. Ann. Math. 106, 93–100 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  10. De Carli, L., Hudson, S.M.: A Faber–Krahn inequality for solutions of Schrödinger’s equation. Adv. Math. 230, 2416–2427 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Faber, C.: Beweis, dass unter allen homogenen Membrane von gleicher Fläche und gleicher Spannung die kreisförmige die tiefsten Grundton gibt. Sitzungsber. Bayer. Akad. Wiss. Math. Phys. München 169–172 (1923)

  12. Helffer, B., Hoffmann-Ostenhof, T.: A review on large k minimal spectral \(k\)-partitions and Pleijel’s theorem. In: Spectral Theory and Partial Differential Equations, Contemporary of Mathematics, vol. 640, pp .39–57. arXiv:1509.04501 (2015)

  13. Helffer, B., Persson-Sundqvist, M.: On nodal domains in Euclidean balls. Proc. Amer. Math. Soc. 144(11), 4777–4791 (2016)

  14. Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T.: Local properties of solutions of Schrödinger equations. Commun. Partial Differ. Equ. 17(3–4), 491–522 (1992)

    Article  MATH  Google Scholar 

  15. Hörmander, L.: The Weyl calculus of pseudodifferential operators. Commun. Pure Appl. Math. 32, 359–443 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hörmander, L.: On the asymptotic distribution of the eigenvalues of pseudodifferential operators in \(\mathbb{R}^n\). Arkiv för matematik 17(3), 297–313 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. Krahn, E.: Über eine von Rayleigh formulierte minimal Eigenschaft des Kreises. Math. Ann. 94, 97–100 (1925)

    Article  MathSciNet  MATH  Google Scholar 

  18. Leydold, J.: Knotenlinien und Knotengebiete von Eigenfunktionen. Diplom Arbeit, Universität Wien (1989)

  19. Lieb, E.H.: Bounds on the eigenvalues of the Laplace and Schrdinger operators. Bull. Am. Math. Soc. 82, 751–753 (1976)

    Article  MATH  Google Scholar 

  20. Milnor, J.: On the Betti numbers of real varieties. Proc. Am. Math. Soc. 15, 275–280 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mohamed, A.: Comportement asymptotique, avec estimation du reste, des valeurs propres d’une classe d’opérateurs pseudo-différentiels sur \(\mathbb{R}^n\). Math. Nachr. 140, 127–186 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  22. Peetre, J.: A generalization of Courant nodal theorem. Math. Scand. 5, 15–20 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pleijel, Å.: Remarks on Courant’s nodal theorem. Commun. Pure. Appl. Math. 9, 543–550 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  24. Polterovich, I.: Pleijel’s nodal domain theorem for free membranes. Proc. Am. Math. Soc. 137(3), 1021–1024 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Reed, M., Simon, B.: Method of Modern Mathematical Physics IV: Analysis of operators. Academic Press, New York (1978)

    MATH  Google Scholar 

  26. Robert, D.: Propriétés spectrales d’opérateurs pseudo-différentiels. Commun. PDE 3(9), 755–826 (1978)

    Article  MATH  Google Scholar 

  27. Rosenbljum, G.V.: The distribution of the discrete spectrum for singular differential operators. Sov. Math. Dokl. 13, 245–249 (1972)

    Google Scholar 

  28. Shubin, M.A.: Asymptotic behaviour of the spectral function. In: Pseudodifferential Operators and Spectral Theory, pp. 133–173. Springer, Berlin, Heidelberg (2001)

  29. Shubin, M.A., Tulovskii, V.N.: On the asymptotic distribution of the eigenvalues of pseudodifferential operators in \(\mathbb{R}^n\). Mat. Sb. 92(134), 571–588 (1973). (in Russian)

    MathSciNet  MATH  Google Scholar 

  30. Steinerberger, S.: A geometric uncertainty principle with an application to Pleijel’s estimate. Ann. Henri Poincaré 15(12), 2299–2319 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tamura, H.: Asymptotic formulas with remainder estimates for eigenvalues of Schrödinger operators. Commun. Partial Differ. Equ. 7(1), 1–53 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  32. Tamura, H.: Asymptotic formulas with sharp remainder estimates for bound states of Schrödinger I. J. Anal. Math. 40, 166–182 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tamura, H.: Asymptotic formulas with sharp remainder estimates for bound states of Schrödinger II. J. Anal. Math. 41, 85–108 (1982)

    Article  MATH  Google Scholar 

  34. Weyl, H.: Über die asymptotische Verteilung der Eigenwerte. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen 110–117 (1911)

Download references

Acknowledgements

Thanks to the ESI where the paper was initiated (B. H. and T. H.-O.). The authors would also thank I. Polterovich for helpful discussions at various stages of this work.

Author information

Authors and Affiliations

  1. Université de Montréal, 2920, Chemin de la Tour, Montréal, QC, H3T 1J4, Canada

    Philippe Charron

  2. Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2 rue de la Houssinière, 44322, Nantes, France

    Bernard Helffer

  3. Laboratoire de Mathématiques, Université Paris-Sud, CNRS, Univ. Paris Saclay, Paris, France

    Bernard Helffer

  4. Department of Theoretical Chemistry, Vienna University, Währingerstrasse 17, 1090, Vienna, Austria

    Thomas Hoffmann-Ostenhof

Authors
  1. Philippe Charron
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernard Helffer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Hoffmann-Ostenhof
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philippe Charron.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Charron, P., Helffer, B. & Hoffmann-Ostenhof, T. Pleijel’s theorem for Schrödinger operators with radial potentials. Ann. Math. Québec 42, 7–29 (2018). https://doi.org/10.1007/s40316-017-0078-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40316-017-0078-x

Keywords

Mathematics Subject Classification

Search

Navigation