Letters in Mathematical Physics

, Volume 29, Issue 2, pp 133–150 | Cite as

The critical temperature and gap solution in the Bardeen-Cooper-Schrieffer theory of superconductivity

  • Qiang Du
  • Yisong Yang


The Letter studies the problem of numerical approximations of the critical transition temperature and the energy gap function in the Bardeen-Cooper-Schrieffer equation arising in superconductivity theory. The positive kernel function leads to a phonon-dominant state at zero temperature. Much attention is paid to the equation defined on a bounded region. Two discretized versions of the equation are introduced. The first version approximates the desired solution from below, while the second, from above. Numerical examples are presented to illustrate the efficiency of the method. Besides, the approximations of a full space solution and the associated critical temperature by solution sequences constructed on bounded domains are also investigated.

Mathematics Subject Classifications (1991)

82B26 83D55 45G10 45L 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Billard, P. and Fano, G., An existence proof for the gap equation in the super-conductivity theory,Comm. Math. Phys. 10, 274–279 (1968).Google Scholar
  2. 2.
    Davydov, A. S., Theoretical investigation of high-temperature superconductivity,Phys. Rep. 190, 191–306 (1990).Google Scholar
  3. 3.
    Du, Y., Uniqueness results for the gap equation in the theory of superconductivity, Preprint, 1992.Google Scholar
  4. 4.
    Enz, C. P., Introduction to the physics of high-temperature superconductors,Helv. Phys. Acta 61, 741–759 (1988).Google Scholar
  5. 5.
    Hugenholtz, N. M., Quantum theory of many-body systems,Rep. Progr. Phys. 28, 201–247 (1965).Google Scholar
  6. 6.
    Van Hemmen, L., Linear fermion systems, molecular field models, and the KMS condition,Fortschr. Phys. 26, 379–439 (1978).Google Scholar
  7. 7.
    Kitamura, M., Nonlinear integral equation of the Hammerstein type,Progr. Theoret. Phys. 30, 435–442 (1963).Google Scholar
  8. 8.
    Weger, M. and Englman, R., A dynamic Jahn-Teller theory for high-T c superconductivity,Physica A 168, 324–337 (1990).Google Scholar
  9. 9.
    Weger, M., Engleman, R., and Halperin, B., Oscillatory solutions of the BCS gap equation,Solid State Comm. 71, 17–72 (1988).Google Scholar
  10. 10.
    Yang, Y., On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity,Lett. Math. Phys. 22, 27–37 (1991).Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Qiang Du
    • 1
  • Yisong Yang
    • 2
  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Department of MathematicsCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations