Nonlinear integral equation from the BCS gap equations of superconductivity. (English) Zbl 1183.82097

Nee, Janpou

Nonlinear Anal., Real World Appl. 11, No. 1, 190-197 (2010).

The main result of the present paper establishes that the critical temperature \(T_c\) of the occurence of superconductivity is determined by the the first eigenvalue of the kernel function in the context of the Markowitz-Kadanoff model. The results developed in the present paper show that the determination of the critical temperature may be reduced to a singleton equation. The proofs rely on elementary monotonicity techniques.

Reviewer: Vicenţiu D. Rădulescu (Craiova)

MSC:

82D55	Statistical mechanical studies of superconductors
45G15	Systems of nonlinear integral equations

Keywords:

existence and uniqueness of positive solutions; eigenvalue; system of integral equation; superconductivity; Bardeen-Cooper-Schieffer (BCS) theory

References:

[1]	Allen, P.B.; Mitrovic, B., Theory of superconducting \(T_c\), Solid state phys., 37, 1-92, (1982)
[2]	Bardeen, J.; Cooper, L.N.; Schrieffer, J.R., Theory of superconductivity, Phys. rev., 108, 1175-1204, (1957) · Zbl 0090.45401
[3]	Cooper, L.N., Bound electron pairs in degenerate Fermi gas, Phys. rev., 104, 1189-1190, (1956) · Zbl 0074.23705
[4]	Du, Q.; Yang, Y., The critical temperature and gap solution in the bardeen – cooper – schrieffer theory of superconductivity, Lett. math. phys., 29, 133-150, (1993) · Zbl 0787.65105
[5]	Geilikman, B.T.; Zaitsev, R.O.; Kresin, V.Z., Properties of superconductors having overlapping bands, Sov. phys.—solid state, 9, 642-647, (1967)
[6]	Ketterson, J.B.; Song, S.N., Superconductivity, (1999), Cambridge U. Press Cambridge
[7]	Leggett, A.J., Number-phase fluctuations in two-band superconductors, Progr. theoret. phys., 36, 901-930, (1966)
[8]	London, F.; London, H., Proc. R. soc. London, ser. A, 149, 71, (1935)
[9]	Maxwell, E., Isotope effect in the superconductivity of Mercury, Phys. rev., 78, 477, (1950)
[10]	Meissner, W.; Ochsenfeld, R., Naturwissenschaften, 21, 787, (1933)
[11]	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) · Zbl 1054.82036
[12]	Rickayzen, G., Theory of superconductivity, (1965), Interscience New York · Zbl 0138.22703
[13]	Schrieffer, J.R., Theory of superconductivity, (1988), Addison-Wesley New York · Zbl 0125.24102
[14]	Suhl, H.; Matthias, B.T.; Walker, L.R., Bardeen – cooper – schrieffer theory of superconductivity in the case of overlapping bands, Phys. rev. lett., 3, 552-554, (1959) · Zbl 0088.45601
[15]	Tilley, D.R.; Tilley, J., Superfluidity and superconductivity, (1990), IOP Pub. New York
[16]	Tinkham, M., Introduction to superconductivity, (1996), McGraw-Hill New York
[17]	Yang, Y., On the bardeen – cooper – schrieffer integral equation in the theory of superconductivity, Lett. math. phys., 22, 27-37, (1991) · Zbl 0729.45009
[18]	Yang, Y., A mathematical analysis of multiband BCS gap equations in superconductivity, Phys. D, 200, 1-2, 60-74, (2005) · Zbl 1062.82060