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Integrable superconductivity and Richardson equations

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Abstract

For the integrable generalized model of superconductivity, the solution of the Richardson equations is studied for a model spectrum. For the case of a narrow band, the solution is presented in terms of generalized Laguerre or Jacobi polynomials. In the asymptotic limit, when the Richardson equations are transformed into a singular integral equation, the properties of the integration contour are discussed and the spectral density is calculated. The conditions of appearance of gaps in the spectrum are investigated.

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References

  1. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev., 108, 1175–1204 (1957).

    Article  MATH  Google Scholar 

  2. N. N. Boholyubov, D. N. Zubarev, and Yu. A. Tserkovnikov, “On the theory of phase transitions,” Dokl. Akad. Nauk SSSR, 117, No. 5, 788–791 (1957).

    Google Scholar 

  3. R. W. Richardson, “A restricted class of exact eigenstates of the pairing-force Hamiltonian,” Phys. Lett., 3, No. 6, 277–279 (1963).

    Article  MATH  Google Scholar 

  4. M. Gaudin, “Diagonalization d’une class d’hamiltoniens de spin,” J. Phys., 37, No. 10, 1087–1098 (1976).

    Google Scholar 

  5. M. Gaudin, La Fonction d’Onde de Bethe, Masson, Paris (1983).

    MATH  Google Scholar 

  6. M. Gaudin, “Modèles exactement résolus,” CEA Saclay-Service Phys. Théor. Edit. Phys. (1995).

  7. J. Dukelsky, S. Pittel, and G. Sierra, “Exactly solvable Richardson-Gaudin models for many-body quantum systems,” Nuclph/0405011.

  8. D. Hilbert, “Über die Discriminante der im Endlichen abbrechenden hypergeometrischen Reihe,” J. Math., 103, 337–345 (1888).

    Google Scholar 

  9. G. Szego, Orthogonal Polynomials, Amer. Math. Soc., New York (1959).

    Google Scholar 

  10. E. Hille, Ordinary Differential Equations in the Complex Domain, John Wiley & Sons, New York (1976).

    MATH  Google Scholar 

  11. E. L. Ince, Ordinary Differential Equations, Dover Publ., New York (1956).

    Google Scholar 

  12. F. D. Gakhov, Boundary-Value Problems [in Russian], Nauka, Moscow (1977).

    MATH  Google Scholar 

  13. N. I. Muskhelishvili, Singular Integral Equations [in Russian], Nauka, Moscow (512).

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Authors and Affiliations

  1. Institute of Magnetism, Ukrainian National Academy of Sciences, Kyiv

    E. D. Belokolos

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  1. E. D. Belokolos
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 314–326, March, 2007.

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Belokolos, E.D. Integrable superconductivity and Richardson equations. Ukr Math J 59, 343–360 (2007). https://doi.org/10.1007/s11253-007-0022-3

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  • DOI: https://doi.org/10.1007/s11253-007-0022-3

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