Advertisement

The N7/5 Law for Charged Bosons

  • Joseph G. Conlon
  • Elliott H. Lieb
  • Horng-Tzer Yau

Abstract

Non-relativistic bosons interacting with Coulomb forces are unstable, as Dyson showed 20 years ago, in the sense that the ground state energy satisfies E 0 ≦ -AN 7/5. We prove that 7/5 is the correct power by proving that E 0 ≧ -BN 7/5. For the non-relativistic bosonic, one-component jellium problem, Foldy and Girardeau showed that E 0 ≦-CNρ 1/4. This 1/4 law is also validated here by showing that E 0 ≧ -DNρ 1/4. These bounds prove that the Bogoliubov type paired wave function correctly predicts the order of magnitude of the energy.

Keywords

Kinetic Energy Thermodynamic Limit Coulomb Potential Yukawa Potential Neutral Case 
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.

References

  1. 1.
    Bogoliubov, N. N.: On the theory of superfluidity. J. Phys. (USSR) 11, 23–32 (1947)Google Scholar
  2. 2.
    Conlon, J. G.: The ground state energy of a Bose gas with Coulomb interaction II. Commun. Math. Phys. 108, 363–374 (1987). See also part I, ibid 100, 355–397 (1985)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Conlon, J. G.: The ground state energy of a classical gas. Commun. Math. Phys. 94, 439–458 (1984)MathSciNetADSCrossRefMATHGoogle Scholar
  4. 4.
    Dyson, F. J.: Ground-state energy of a finite system of charged particles. J. Math. Phys. 8, 1538–1545 (1967)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Dyson, F. J., Lenard, A.: Stability of matter I and II. J. Math. Phys. 8, 423–434 (1967); ibid 9, 698–711 (1968)MathSciNetADSCrossRefMATHGoogle Scholar
  6. 6.
    Federbush, P.: A new approach to the stability of matter problem II. J. Math. Phys. 16, 706–709 (1975)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Foldy, L. L.: Charged boson gas. Phys. Rev. 124, 649–651 (1961); Errata ibid 125, 2208 (1962)ADSCrossRefGoogle Scholar
  8. 8.
    Girardeau, M.: Ground state of the charged Bose gas. Phys. Rev. 127, 1809–1818 (1962)ADSCrossRefMATHGoogle Scholar
  9. 9.
    Lieb, E. H.: The N 5/3 law for bosons. Phys. Lett. 70A, 71–73 (1979)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Lieb, E. H.: The Bose fluid. In: Lectures in Theoretical Physics, Vol. VII C, pp. 175–224. Brittin, W. E. (ed.). Boulder: University of Colorado Press 1965Google Scholar
  11. 11.
    Lieb, E. H.: On characteristic exponents in turbulence. Commun. Math. Phys. 92, 473–480 (1984)MathSciNetADSCrossRefMATHGoogle Scholar
  12. 12.
    Lieb, E. H., Narnhofer, H.: The thermodynamic limit for jellium. J. Stat. Phys. 12, 291–310 (1975). Errata, ibid 14, 465 (1976)MathSciNetADSCrossRefMATHGoogle Scholar
  13. 13.
    Lieb, E. H., Sakakura, A. Y.: Simplified approach to the ground-state energy of an imperfect Bose gas II. Charged Bose gas at high density. Phys. Rev. 133, A899-A906 (1964)ADSCrossRefGoogle Scholar
  14. 14.
    Lieb, E. H., Simon, B.: The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977). See also, Lieb, E. H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603–641 (1981); Errata ibid 54, 311 (1982)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lieb, E. H., Thirring, W. E.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687–689 (1975). Errata, ibid 35, 1116 (1975).ADSCrossRefGoogle Scholar
  16. 16.
    Thirring, W. E.: A course in mathematical physics, Vol. 4. Quantum mechanics of large systems. New York, Wien: Springer 1983CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Joseph G. Conlon
    • 1
  • Elliott H. Lieb
    • 2
  • Horng-Tzer Yau
    • 2
  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

Personalised recommendations