Skip to main content
Log in

A lower bound for the number of negative eigenvalues on a Euclidean space and on a complete Riemannian manifold

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

The purpose of the present manuscript is to provide a lower bound for the number of negative eigenvalues associated to a generalized Schrödinger operator, this lower bound is given in terms of a finite number of cubes.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. El Aïdi, M.: Sur le nombre des valeurs propres négatives d’un opérateur elliptique. Bull. Sci. math. 137, 434–456 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. El Aïdi, M.: Un majorant du nombre des valeurs propres négatives correspondantes à l’opérateur de Schrödinger généralisé. Annales mathématiques Blaise Pascal 19, 197–211 (2012)

    Article  MATH  Google Scholar 

  3. Maz’ya, V.G.: Sobolev space with Applications to Elliptic Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, Vol. 342, 2nd augmented Edition (2011)

  4. Verbitsky, I.: Nonlinear potentials and trace inequalities, The Maz’ya anniversary collection, Vol. 2 (Rostock, 1998). Oper. Theory Adv. Appl. 110: 323–343 (1999)

  5. Egorov, Y., El Aïdi, M.: Spectre négatif d’un opérateur elliptique avec des conditions au bords de Robin. Publ. Mat. 45, 125–148 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Egorov, Y.V., Kondratiev, V.A.: Estimates of the negative spectrum of an elliptic operator, in Spectral theory of operators (Novgorod, 1989). Am. Math. Soc. Transl. Ser. 2 150, 111–140 (1989)

    Google Scholar 

  7. Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106(2), 93–100 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  8. Lieb, E.H.: Bounds on the eigenvalues of the Laplace and Schrdinger operators. Bull. Am. Math. Soc. 82(5), 751–753 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  9. Rozenbljum, G.V.: Distribution of the discrete spectrum of singular differential operators. Dokl. Akad. Nauk. SSSR 202, 1012–1015 (1972) (in Russian)

  10. El Aïdi, M.: On a new embedding theorem and the CLR-type inequality for Euclidean and hyperbolic spaces. Bull. Sci. Math. 138, 335–342 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kerman et, R., Sawyer, E.T.: The trace inequality and eigenvalue estimates for Shrödinger operators. Ann. Inst. Fourier Grenoble 4(36), 207–228 (1986)

    Article  Google Scholar 

  12. Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  13. Lee, J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics 218. Springer, Berlin (2003)

    Book  Google Scholar 

  14. Simon, B., Reed, M.: Methods of Mathematical Physics I: Functional Analysis. Academic Press, Inc., New York (1978)

    Google Scholar 

  15. Glazman, I.M.: Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Gosudarstv. Izdat. Fiz.-Mat. Lit. Moscow (1963) (English translation: Daniel Davey and Co., New York, 1966)

  16. Shubin, M.A.: Pseudodifferential Operators and Spectral Theory, 2nd edn. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  17. Stein, E.M.: Singular Integrals Differentiability Properties of Functions. Princeton University Press, USA (1970)

    MATH  Google Scholar 

  18. Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Grundlehren der mathematischen Wissenschaften, Vol. 314, Springer, Berlin (1999) (corrected second printing)

  19. Ziemer, W.P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  20. Jürgen, J.: Riemannian Geometry and Geometric Analysis, 4th edn. Springer, Berlin (2005)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mohammed El Aïdi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El Aïdi, M. A lower bound for the number of negative eigenvalues on a Euclidean space and on a complete Riemannian manifold. J. Pseudo-Differ. Oper. Appl. 5, 481–490 (2014). https://doi.org/10.1007/s11868-014-0098-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11868-014-0098-0

Keywords

Mathematics Subject Classification

Navigation