Inequalities pp 239-241 | Cite as

On Semi-Classical Bounds for Eigenvalues of Schrödinger Operators

  • Michael Aizenman
  • Elliott H. Lieb


Our principal result is that if the semiclassical estimate is a bound for some moment of the negative eigenvalues (as is known in some cases in one-dimension), then the semiclassical estimates are also bounds for all higher moments.


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  1. [1]
    E.H. Lieb and W.E. Thirring, Phys. Rev. Lett. 35 (1975) 687. See Phys. Rev. Lett. 35 (1975) 1116 for errata. Also E.H. Lieb, Rev. Mod. Phys. 48 (1976) 553.CrossRefGoogle Scholar
  2. [2]
    E.H. Lieb and W.E. Thirring, in: Studies in mathematical physics, Essays in honor of V. Bargmann (Princeton Univ. Press, Princeton, N.J., 1976).Google Scholar
  3. [3]
    A. Martin, Helv. Phys. Acta 45 (1972) 140.Google Scholar
  4. [4]
    H. Tamura, Proc. Japan Acad. 50 (1974) 19.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    M. Cwikel, Ann. of Math. 106 (1977) 93.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    E.H. Lieb, Bull. Amer. Math. Soc. 82 (1976) 751.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    V. Glaser, H. Grosse and A. Martin, Bounds on the Number of Eigenvalues of the Schrödinger Operator, CERN preprint TH2432 (1977).Google Scholar
  8. [8]
    G.V. Rosenblum, The distribution of the discrete spectrum for singular differential operators, Isvestia Math. 164 No. 1 (1976) 75.Google Scholar
  9. [9]
    M.S. Birman and V.V. Borzov, On the asymptotics of the discrete spectrum of some singular differential operators, Topics in Math. Phys. 5 (1972) 19.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Michael Aizenman
    • 1
  • Elliott H. Lieb
    • 2
  1. 1.Department of PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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