Abstract
The BCS Hamiltonian of superconductivity has the second branch of eigenvalues and eigenvectors. It consists of wave functions of pairs of electrons in ground and excited states. The continuous spectrum of excited pairs is separated by a nonzero gap from the point of the discrete spectrum that corresponds to the pair in the ground state. The corresponding grand partition function and free energy are exactly calculated. This implies that, for low temperatures, the system is in the condensate of pairs in the ground state. The sequence of correlation functions is exactly calculated in the thermodynamic limit, and it coincides with the corresponding sequence of the system with approximating Hamiltonian. The gap in the spectrum of excitations depends continuously on temperature and is different from zero above the critical temperature corresponding to the first branch of the spectrum. In our opinion, this fact explains the phenomenon of “pseudogap.”
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References
D. Ya. Petrina, “Spectrum and states of the BCS Hamiltonian in a finite domain. I. Spectrum,” Ukr. Mat. Zh., 52, No. 5, 667–689 (2000).
D. Ya. Petrina, “Spectrum and states of the BCS Hamiltonian in a finite domain. II. Spectra of excitations,” Ukr. Mat. Zh., 53, No. 8, 1080–1100 (2001).
D. Ya. Petrina, “Spectrum and states of the BCS Hamiltonian in a finite domain. III. The BCS Hamiltonian with mean-field interaction,” Ukr. Mat. Zh., 54, No. 11, 1486–1504 (2002).
D. Ya. Petrina, “Model BCS Hamiltonian and approximating Hamiltonian in the case of infinite volume. IV. Two branches of their common spectra and states,” Ukr. Mat. Zh., 55, No. 2, 174–196 (2003).
D. Ya. Petrina, “BCS model Hamiltonian of the theory of superconductivity as a quadratic form,” Ukr. Mat. Zh., 56, No. 3, 309–338 (2004).
D. Ya. Petrina, “On Hamiltonians of quantum statistical mechanics and a model Hamiltonian in the theory of superconductivity,” Teor. Mat. Fiz., 4, No. 3, 394–411 (1970).
D. Ya. Petrina and V. P. Yatsyshin, “On a model Hamiltonian in the theory of superconductivity,” Teor. Mat. Fiz., 10, No. 2, 283–299 (1972).
J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev., 108, No. 5, 1175–1204 (1957).
N. N. Bogolyubov, “On a model Hamiltonian in the theory of superconductivity,” in: N. N. Bogolyubov, Selected Papers [in Russian], Vol. 3, Naukova Dumka, Kiev (1970), pp. 110–173.
L. N. Cooper, “Bound electron pairs in a degenerate Fermi gas,” Phys. Rev., 104, No. 4, 1189–1190 (1956).
Y. Yamaguchi, “Two-nucleon problem where the potential is nonlocal but separable. I, II,” Phys. Rev., 95, No. 6, 1628–1634, 1635–1643 (1954).
R. G. Newton, Scattering Theory of Waves and Particles [Russian translation], Mir, Moscow (1969).
G. N. Watson, Bessel Functions, I, New York (1958).
M. R. Schafroth, S. T. Butler, and J. M. Blatt, Helv. Phys. Acta, 30, 93 (1957).
D. Ya. Petrina, Mathematical Foundations of Quantum Statistical Mechanics, Kluwer, Dordrecht (1995).
N. N. Bogolyubov, Jr., A Method for the Investigation of a Model Hamiltonian [in Russian], Nauka, Moscow (1974).
J. R. Schrieffer, Theory of Superconductivity [Russian translation], Nauka, Moscow (1970).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1508–1533, November, 2005.
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Petrina, D.Y. New second branch of the spectrum of the BCS Hamiltonian and a “pseudogap”. Ukr Math J 57, 1763–1791 (2005). https://doi.org/10.1007/s11253-006-0028-2
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DOI: https://doi.org/10.1007/s11253-006-0028-2