Integrable superconductivity and Richardson equations

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