Communications in Mathematical Physics

, Volume 252, Issue 1–3, pp 485–534 | Cite as

Ground State Energy of the Two-Component Charged Bose Gas

  • Elliott H. Lieb
  • Jan Philip SolovejEmail author


We continue the study of the two-component charged Bose gas initiated by Dyson in 1967. He showed that the ground state energy for N particles is at least as negative as −CN7/5 for large N and this power law was verified by a lower bound found by Conlon, Lieb and Yau in 1988. Dyson conjectured that the exact constant C was given by a mean-field minimization problem that used, as input, Foldy’s calculation (using Bogolubov’s 1947 formalism) for the one-component gas. Earlier we showed that Foldy’s calculation is exact insofar as a lower bound of his form was obtained. In this paper we do the same thing for Dyson’s conjecture. The two-component case is considerably more difficult because the gas is very non-homogeneous in its ground state.


Neural Network Statistical Physic Complex System State Energy Nonlinear Dynamics 
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  1. 1.
    Benguria, R.D.: The Lane-Emden equation revisited. In: Advances in differential equations and mathematical physics (Birmingham, AL, 2002), Contemp. Math. 327, Providence, RI: Am. Math. Soc., 2003, pp. 11–19Google Scholar
  2. 2.
    Bogolubov, N.N.: J. Phys. (U.S.S.R.) 11, 23 (1947); Bogolubov, N.N., Zubarev, D.N. Sov. Phys. JETP 1, 83 (1955)Google Scholar
  3. 3.
    Conlon, J.G., Lieb, E.H., Yau, H-T.: The N7/5 law for charged bosons. Commun. Math. Phys. 116, 417–448 (1988)MathSciNetGoogle Scholar
  4. 4.
    Dyson, F.J.: Ground-state energy of a finite system of charged particles. J. Math. Phys. 8, 1538–1545 (1967)Google Scholar
  5. 5.
    Foldy, L.L.: Charged boson gas, Phys. Rev. 124, 649–651 (1961); Errata ibid 125, 2208 (1962)CrossRefzbMATHGoogle Scholar
  6. 6.
    Kwong, M.K.: Uniqueness of positive solutions of Δu-u+up=0 in Rn. Arch. Rat. Mech. Anal. 105(3), 243–266 (1989)zbMATHGoogle Scholar
  7. 7.
    Lieb, E.H.: The Bose fluid. In: Brittin, W.E. (eds.), Lecture Notes in Theoretical Physics VIIC, Boulder, Co: Univ. of Colorado Press, 1964, pp. 175–224Google Scholar
  8. 8.
    Lieb, E.H., Solovej, J.P.: Ground State Energy of the One-Component Charged Bose Gas. Commun. Math. Phys. 217 (1), 127–163 (2001) (Erratum: Commun. Math. Phys. 225, 219–221 (2002))CrossRefzbMATHGoogle Scholar
  9. 9.
    McLeod, K., Serrin, J.: Uniqueness of positive radial solutions of Δu+f(u)=0 in Rn. Arch. Rat. Mech. Anal. 99(2), 115–145 (1987)zbMATHGoogle Scholar
  10. 10.
    Onsager, L.: Electrostatic Interaction of Molecules. J. Phys. Chem. 43, 189–196 (1939)Google Scholar
  11. 11.
    Solovej, J.P.: Upper Bounds to the Ground State Energies of the One- and Two-Component Charged Bose Gases. arXiv math-ph/0406014Google Scholar
  12. 12.
    Zhang, L.Q.: Uniqueness of ground state solutions. Acta Math. Sci. (English Ed.) 8(4), 449–467 (1988)zbMATHGoogle Scholar

Copyright information

© E.H. Lieb and J.P. Solovej 2004

Authors and Affiliations

  1. 1.Departments of Physics and MathematicsJadwin Hall, Princeton UniversityPrincetonUSA
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA

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