Monodromy matrix for linear difference operators with almost constant coefficients (Q1910847)

A new method for solving the direct scattering problem associated with a linear difference operator of arbitrary order \[ L = \widetilde p_{-k} (t) E^{-k} + \widetilde p_{- k + 1} (t) E^{- k + 1} + \cdots + \widetilde p_{m - 1} (t) E^{m - 1} + \widetilde p_m (t) E^m, \] is presented. A procedure for analysing the behavior of the monodromy operator (monodromy matrix) under the influence of shift is also presented. Finally it is proved that the second-order self adjoint difference operator (corresponding to the discrete Schrödinger scattering problem) associated with the Toda lattice equations possesses the \(l^2\)-eigenfunctions.