Abstract
We prove upper bounds on the ground state energies of the one- and two-component charged Bose gases. The upper bound for the one-component gas agrees with the high density asymptotic formula proposed by L. Foldy in 1961. The upper bound for the two-component gas agrees in the large particle number limit with the asymptotic formula conjectured by F. Dyson in 1967. Matching asymptotic lower bounds for these systems were proved in references [10] and [11]. The formulas of Foldy and Dyson which are based on Bogolubov’s pairing theory have thus been validated.
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References
Bach V., Lieb E.H., Solovej J.P. (1994) Generalized Hartree-Fock theory and the Hubbard model. J. Stat. Phys. 76, 3–90
Berezin, F.A. Izv. Akad. Nauk, ser. mat. 36(No. 5) (1972); English translation: USSR Izv. 6(No. 5) (1972); Berezin, F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975)
Bhatia R. (1997) Matrix Analysis, Graduate Texts in Mathematics, Vol. 169. Springer-Verlag, New York
Bogolubov, N.N. J. Phys. (U.S.S.R.) 11, 23 (1947); Bogolubov, N.N., Zubarev, D.N.: Sov. Phys.-JETP 1, 83 (1955)
Conlon J., Lieb E.H., Yau H.-T. (1988) The N 7/5 Law for Charged Bosons. Commun. Math. Phys. 116, 417–448
Dyson F.J. (1967) Ground State Energy of a Finite System of Charged Particles. J. Math. Phys. 8, 1538–1545
Foldy, L.L. (1973) Charged Boson Gas. Phys. Rev. 124, 649–651 (1961); Errata ibid 125, 2208 (1962)
Lieb E.H. (1973) The classical limit of quantum spin systems. Commun. Math. Phys. 31, 327–340
Lieb, E.H., Narnhofer, H. The Thermodynamic Limit for Jellium. J. Stat. Phys. 12, 291–310 (1975); Errata 14, 465 (1976)
Lieb, E.H., Solovej, J.P. Ground State Energy of the One-Component Charged Bose Gas. Commun. Math. Phys. 217, 127–163 (2001). Errata 225, 219–221 (2002)
Lieb E.H., Solovej J.P. (2004) Ground State Energy of the Two-Component Charged Bose Gas. Commun. Math. Phys. 252, 485–534
Robinson D.W. (1965) The ground state of the Bose gas. Commun. Math. Phys. 1, 159–174
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Communicated by M. Aizenman
© 2006 by the author. This article may be reproduced in its entirety for non-commercial purposes.
Work partially supported by NSF grant DMS-0111298, by EU grant HPRN-CT-2002-00277, by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation, and by grants from the Danish research council.
Most of this work was done while the author was visiting the School of Mathematics, Institute for Advanced Study, Princeton.
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Solovej, J.P. Upper Bounds to the Ground State Energies of the One- and Two-Component Charged Bose Gases. Commun. Math. Phys. 266, 797–818 (2006). https://doi.org/10.1007/s00220-006-0020-6
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DOI: https://doi.org/10.1007/s00220-006-0020-6