Eigenvalue problem for tridiagonal matrices arising in the scattering-theory analysis of disordered conductors

Abstract

The eigenvalue problem for a family of tridiagonal matrices arising in the scattering-theory analysis of the conductance in mesoscopic systems, and its fluctuations, is studied. The exactly solvable special cases are identified. For the general problem, qualitative characteristics of the spectrum are established, and approximate solutions for the eigenvalues are constructed. These comprise ones that are valid in the limit of large but finite matrix dimension, and those derived from a perturbation expansion around each of the exactly solvable cases.