Monodromy Matrix for Linear Difference Operators with Almost Constant Coefficients

A new method for solving the discrete scattering problem for a linear single-valued difference operator of arbitrary order with almost constant coefficients is proposed. The treatment is concerned with the asymptotic behavior of its eigenfunctions as |t| → ∞. The purpose of the paper is to investigate the transition between the asymptotically free states of the underlying system, defined in terms of the monodromy operator. A rare mathematical tool is used: an infinite matrix product. It is shown that, if the coefficients of the aforementioned operator are suitably behaved, the monodromy operator exists in the form of the convergent two-sided infinite product of matrices associated with the matrix eigenvalue equation corresponding to the scattering problem. The application of the method to the solution of the scattering problem for Toda lattice equations is demonstrated. However, the approach is quite general and should be applicable to other forms of lattice equations.