Communications in Mathematical Physics

, Volume 281, Issue 2, pp 349–367 | Cite as

The BCS Functional for General Pair Interactions

  • Christian HainzlEmail author
  • Eman Hamza
  • Robert Seiringer
  • Jan Philip Solovej


The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.


Critical Temperature Negative Eigenvalue Interaction Potential Versus Canonical Anticommutation Relation General Pair Interaction 
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  1. 1.
    Andrenacci N., Perali A., Pieri P., Strinati G.C.: Density-induced BCS to Bose-Einstein crossover. Phys. Rev. B 60, 12410 (1999)CrossRefGoogle Scholar
  2. 2.
    Bach V., Lieb E., Solovej J.: Generalized Hartree-Fock theory and the Hubbard model. J. Stat. Phys. 76, 3–89 (1994)CrossRefADSMathSciNetzbMATHGoogle Scholar
  3. 3.
    Bardeen J., Cooper L., Schrieffer J.: Theory of Superconductivity. Phys. Rev. 108, 1175–1204 (1957)CrossRefADSMathSciNetzbMATHGoogle Scholar
  4. 4.
    Billard P., Fano G.: An existence proof for the gap equation in the superconductivity theory. Commun. Math. Phys. 10, 274–279 (1968)zbMATHGoogle Scholar
  5. 5.
    Bloch, I., Dalibard, J., Zwerger, W.: Many-Body Physics with Ultracold Gases., 2007, to appear in Rev. Mod. Phys.
  6. 6.
    Carlson J., Chang S.-Y., Pandharipande V.R., Schmidt K.E.: Superfluid Fermi Gases with Large Scattering Length. Phys. Rev. Lett. 91, 0504011 (2003)Google Scholar
  7. 7.
    Chen Q., Stajic J., Tan S., Levin K.: BCS–BEC crossover: From high temperature superconductors to ultracold superfluids. Phys. Rep. 412, 1–88 (2005)CrossRefADSGoogle Scholar
  8. 8.
    Fetter A., Walecka J.D.: Quantum theory of many-particle systems. McGraw-Hill, New-York (1971)Google Scholar
  9. 9.
    Frank R.L., Hainzl C., Naboko S., Seiringer R.: The critical temperature for the BCS equation at weak coupling. J. Geom. Anal. 17, 559–568 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Leggett, A.J.: Diatomic Molecules and Cooper Pairs. Modern trends in the theory of condensed matter, J. Phys. (Paris) Colloq, C7–19 Bertin-Heidelberg-New York: Springer, 1980Google Scholar
  11. 11.
    Lieb, E., Loss, M.: Analysis. Providence RI: Amer. Math. Soc., 2001Google Scholar
  12. 12.
    Martin P.A., Rothen F.: Many-body problems and Quantum Field Theory. Springer, Berlin-Heidelberg-New York (2004)zbMATHGoogle Scholar
  13. 13.
    McLeod J.B., Yang Y.: The uniqueness and approximation of a positive solution of the Bardeen-Cooper-Schrieffer gap equation. J. Math. Phys. 41, 6007–6025 (2000)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Nozières P., Schmitt-Rink S.: Bose Condensation in an Attractive Fermion Gas: From Weak to Strong Coupling Superconductivity. J. Low Temp. Phys. 59, 195–211 (1985)Google Scholar
  15. 15.
    Parish M., Mihaila B., Timmermans E., Blagoev K., Littlewood P.: BCS-BEC crossover with a finite-range interaction. Phys. Rev. B 71, 0645131–0645136 (2005)CrossRefGoogle Scholar
  16. 16.
    Randeria, M.: In: Bose-Einstein Condensation, Griffin, A., Snoke, D.W., Stringari, S. eds., Cambridge: Cambridge University Press, 1995Google Scholar
  17. 17.
    Tiesinga E., Verhaar B.J., Stoof H.T.C.: Threshold and resonance phenomena in ultracold ground-state collisions. Phys. Rev. A 47, 4114 (1993)CrossRefGoogle Scholar
  18. 18.
    Vansevenant A.: The gap equation in superconductivity theory. Physica 17D, 339–344 (1985)ADSMathSciNetGoogle Scholar
  19. 19.
    Yang Y.: On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity. Lett. Math. Phys. 22, 27–37 (1991)CrossRefADSMathSciNetzbMATHGoogle Scholar

Copyright information

© The Authors 2008

Authors and Affiliations

  • Christian Hainzl
    • 1
    Email author
  • Eman Hamza
    • 2
  • Robert Seiringer
    • 3
  • Jan Philip Solovej
    • 4
  1. 1.Departments of Mathematics and PhysicsUABBirminghamUSA
  2. 2.Department of MathematicsUABBirminghamUSA
  3. 3.Department of PhysicsPrinceton UniversityPrincetonUSA
  4. 4.Department of MathematicsUniversity of CopenhagenCopenhagenDenmark

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