The Number of Bound States of One-Body Schroedinger Operators and the Weyl Problem

  • Elliott H. Lieb


If Ñ( Ω, λ) is the number of eigenvalues of A in a domain Ω, in a suitable Riemannian manifold of dimension n, we derive bounds of the form Ñ(Ω, λ)< D λn/2}Ω{ for all Ω, λ , n , Likewise, if Nα(V) is the number of nonpositive eigenvalues of -Δ + V(x) which are ≤ α ≤ 0, then \( {N_\alpha }(V) \leqslant {L_n}{\int\limits_M {\left[ {V - \alpha } \right]} _ - }^{n/2} \) for all α and V and n ≥ 3.


Pure Math Semigroup Property Sharp Constant Weak Type Estimate Classical Constant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Weyl, “Das asymptotische Verteilungsgesetz der Eigenwerte Linearer partieller Differentialgleichungen”, Math. Ann. 71 (1911), 441–469.MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. Kac, “Can one hear the shape of a drum?”, Slaught Memorial Papers, no. 11, Amer. Math. Monthly 73 (1966), no. 4, part II, 1–23.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Acad. Press, N. Y., 1978.MATHGoogle Scholar
  4. 4.
    G. V. Rosenbljum, “Distribution of the discrete spectrum of singular differential operators”, Dokl. Aka. Nauk SSSR, 202 (1972), 1012–1015 (MR 45 #4216).Google Scholar
  5. 4a.
    G. V. Rosenbljum, The details are given in “Distribution of the discrete spectrum of singular differential operators”, Izv. Vyss. Ucebn. Zaved. Matematika 164 (1976), 75–86.Google Scholar
  6. 4b.
    G. V. Rosenbljum, [English trans. Sov. Math. (Iz. VUZ) 20 (1976), 63–71.]Google Scholar
  7. 5.
    B. Simon, “Weak trace ideals and the number of bound states of Schroedinger operators”, Trans. Amer. Math. Soc. 224 (1976), 367–380.MathSciNetADSMATHGoogle Scholar
  8. 6.
    M. Cwikel, “Weak type estimates for singular values and the number of bound states of Schroedinger operators”, Ann. Math. 106 (1977), 93–100.MathSciNetCrossRefMATHGoogle Scholar
  9. 7.
    E. Lieb, “Bounds on the eigenvalues of the Laplace and Schroedinger operators”, Bull. Amer. Math. Soc. 82 (1976), 751–753.MathSciNetCrossRefMATHGoogle Scholar
  10. 8.
    B. Simon, Functional Integration and Quantum Physics, Academic Press, N. Y., to appear 1979.MATHGoogle Scholar
  11. 9.
    E. Lieb and W. Thirring, “Inequalities for the moments of the eigenvalues of the Schroedinger equation and their relation to Sobolev inequalities”, in Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann (E. Lieb, B. Simon and A. Wightman eds.), Princeton Univ. Press, Princeton, N. J., 1976.Google Scholar
  12. 9.
    E. Lieb and W. Thirring, These ideas were first announced in “Bound for the kinetic energy of fermions which proves the stability of matter”, Phys. Rev. Lett. 35 (1975), 687–689,ADSCrossRefGoogle Scholar
  13. 9.
    E. Lieb and W. Thirring, Errata 35 (1975), 1116.Google Scholar
  14. 10.
    M. Aizenman and E. Lieb, “On semi-classical bounds for eigenvalues of Schroedinger operators”, Phys. Lett. 66A (1978), 427–429.MathSciNetCrossRefGoogle Scholar
  15. 11.
    M. Birman, “The spectrum of singular boundary problems”, Math. Sb. 55 (1961), 124–174.MathSciNetGoogle Scholar
  16. 11a.
    M. Birman, “The spectrum of singular boundary problems”, (Amer. Math. Soc. Trans. 53 (1966), 23–80).MATHGoogle Scholar
  17. 12.
    J. Schwinger, “On the bound states of a given potential”, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 122–129.MathSciNetADSCrossRefGoogle Scholar
  18. 13.
    M. Kac, “On some connections between probability theory and differential and integral equations”, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Univ. of Calif. Press, Berkeley, 1951, 189–215.Google Scholar
  19. 14.
    K. R. Ito, “Estimation of the functional determinants in quantum field theories”, Res. Inst, for Math. Sci., Kyoto Univ. (1979), preprint.Google Scholar
  20. 15.
    E. Lieb, “The stability of matter”, Rev. Mod. Phys. 48 (1976), 553–569.MathSciNetADSCrossRefGoogle Scholar
  21. 16.
    V. Glaser, H. Grosse and A. Martin, “Bounds on the number of eigenvalues of the Schroedinger operator”, Commun. Math. Phys. 59 (1978), 197–212.MathSciNetADSCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  1. 1.Departments of Mathematics and PhysicsPrinceton University Jadwin HallPrincetonUSA

Personalised recommendations