Inequalities pp 203-237 | Cite as

Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and their Relation to Sobolev Inequalities

  • Elliott H. Lieb
  • Walter E. Thirring


Estimates for the number of bound states and their energies, ej ≤ 0, are of obvious importance for the investigation of quantum mechanical Hamiltonians. If the latter are of the single particle form H = ≤ Δ + V(x) in Rn, we shall use available methods to derive the bounds
$${\sum\limits_j {\left| {{e_j}} \right|} ^y} \le {L_{y,n}}\int {{d^n}} x\left| {V(x)} \right|_ - ^{y + n/2},y > \max (0,1 - n/2)$$
Here, \(\left| {V(x)} \right|\_ = - V(x){\rm{ if V(x)}} \le {\rm{0}}\) and is zero otherwise.


Mathematical Physic Sobolev Inequality Bare Matrice Single Particle Form Bound State Eigenvalue 
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  1. [1]
    E. H. Lieb and W. E. Thirring, Phys. Rev. Lett. 35, 687 (1975). See Phys. Rev. Lett. 35, 1116 (1975) for errata.CrossRefGoogle Scholar
  2. [2]
    M. S. Birman, Mat. Sb. 55(97), 125 (1961); Amer. Math. Soc. Translations Ser. 2, 53, 23(1966).MathSciNetGoogle Scholar
  3. [3]
    J. Schwinger, Proc. Nat. Acad. Sci. 47, 122 (1961).MathSciNetCrossRefGoogle Scholar
  4. [4]
    B. Simon, “Quantum Mechanics for Hamiltonians Defined as Quadratic Forms,” Princeton University Press, 1971.Google Scholar
  5. [5]
    B. Simon, “On the Number of Bound States of the Two Body Schrödinger Equation — A Review,” in this volume.Google Scholar
  6. [6]
    A. Martin, Helv. Phys. Acta 45, 140 (1972).Google Scholar
  7. [7]
    H. Tamura, Proc. Japan Acad. 50, 19 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    V. Glaser, A. Martin, H. Grosse and W. Thirring, “A Family of Optimal Conditions for the Absence of Bound States in a Potential,” in this, volume.Google Scholar
  9. [9]
    S. L. Sobolev, Mat. Sb. 46, 471 (1938), in Russian.Google Scholar
  10. [10]
    —, Applications of Functional Analysis in Mathematical Physics, Leningrad (1950), Amer. Math. Soc. Transl, of Monographs, 7 (1963).Google Scholar
  11. [11]
    G. Talenti, Best Constant in Sobolev’s Inequality, Istituto Matematico, Universitá Degli Studi Di Firenze, preprint (1975).Google Scholar
  12. [12]
    G. Rosen, SIAM Jour. Appl. Math. 21, 30 (1971).zbMATHCrossRefGoogle Scholar
  13. [13]
    H. J. Brascamp, E. H. Lieb and J. M. Luttinger, Jour. Funct. Anal. 17, 227 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Commun. Pure and Appl. Math. 27, 97 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    S. A. Moszkowski, Phys. Rev. 89, 474 (1953).CrossRefGoogle Scholar
  16. [16]
    A. E. Green and K. Lee, Phys. Rev. 99, 772 (1955).MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    V. E. Zakharov and L. D. Fadeev, Funkts. Anal, i Ego Pril. 5, 18 (1971). English translation: Funct. Anal, and its Appl. 5, 280 (1971).Google Scholar
  18. [18]
    H. Epstein, Commun. Math. Phys. 31, 317 (1973).zbMATHCrossRefGoogle Scholar
  19. [19]
    E. Seiler and B. Simon, “Bounds in the Yukawa Quantum Field Theory,” Princeton preprint (1975).Google Scholar
  20. [20]
    W. Thirring, T7 Quantenmechanik, Lecture Notes, Institut für Theoretische Physik, University of Vienna.Google Scholar
  21. [21]
    T. Aubin, C. R. Acad. Sc. Paris 280, 279 (1975). The results are stated here without proof; there appears to be a misprint in the expression for Cr,n.zbMATHGoogle Scholar
  22. [22]
    B. Simon, “Weak Trace Ideals and the Bound States of Schrödinger Operators,” Princeton preprint (1975).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  • Walter E. Thirring
    • 2
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Institut für Theoretische PhysikUniversität WienAustria

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