# zbMATH — the first resource for mathematics

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

##### MSC:
 82D55 Statistical mechanical studies of superconductors 45G15 Systems of nonlinear integral equations
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.