Advertisement

Inequalities pp 243-254 | Cite as

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

  • Elliott H. Lieb

Abstract

If N ((Ω,λ) is the number of eigenvalues of -Δ in a domain Ω, in a suitable Riemannian manifold of dimension n, we derive bounds of the form \(\tilde N(\Omega ,\lambda ) \le {D_n}{\lambda ^{n/2}}\left| \Omega \right|\) for all Ω, * , n , Likewise, if N03B1; (V) is the number of nonpositive eigenvalues of -Δ + V (x) which are ≤ a ≤ 0, then \({N_\alpha }(V) \le {L_n}\int {_M} \left[ {V - \alpha } \right]_\_^{n/2}\) for all α and V and n ≥ 3. 1980 Mathematics Subject Classification 35P15.

Keywords

Pure Math Semigroup Property Sharp Constant Schr6dinger Operator Lower Semi Continuous Function 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  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.zbMATHCrossRefGoogle Scholar
  3. 3.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Acad. Press, N. Y., 1978.zbMATHGoogle 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). The details are given in “Distribution of the discrete spectrum of singular differential operators”, Izv. Vyss. Ucebn. Zaved. Matematika 164 (1976), 75-86. [English trans. Sov. Math. (Iz. VUZ) 20 (1976), 63-71.]Google Scholar
  5. 5.
    B. Simon, “Weak trace ideals and the number of bound states of Schroedinger operators”, Trans. Amer. Math. Soc. 224 (1976), 367–380.MathSciNetzbMATHGoogle Scholar
  6. 6.
    M. Cwikel, “Weak type estimates for singular values and the number of bound states of Schroedinger operators”, Ann. Math. 106 (1977), 93–100.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    E. Lieb, “Bounds on the eigenvalues of the Laplace and Schroedinger operators”, Bull. Amer. Math. Soc. 82 (1976), 751–753.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    B. Simon, Functional Integration and Quantum Physics, Academic Press, N. Y., to appear 1979.zbMATHGoogle Scholar
  9. 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. 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, Errata 35 (1975), 1116.Google Scholar
  10. 10.
    M. Aizenman and E. Lieb, “On semi-classical bounds for eigenvalues of Schroedinger operators”, Phys. Lett. 66A (1978), 427–429.MathSciNetGoogle Scholar
  11. 11.
    M. Birman, “The spectrum of singular boundary problems”, Math. Sb. 55 (1961), 124–174. (Amer. Math. Soc. Trans. 53 (1966), 23-80).MathSciNetGoogle Scholar
  12. 12.
    J. Schwinger, “On the bound states of a given potential”, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 122–129.MathSciNetCrossRefGoogle Scholar
  13. 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
  14. 14.
    K. R. Ito, “Estimation of the functional determinants in quantum field theories”, Res. Inst. for Math. Sci., Kyoto Univ. (1979), preprint.Google Scholar
  15. 15.
    E. Lieb, “The stability of matter”, Rev. Mod. Phys. 48 (1976), 553–569.MathSciNetCrossRefGoogle Scholar
  16. 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.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

Personalised recommendations