Inverse scattering problem for a discrete Sturm-Liouville operator on the entire line

The inverse scattering problem for the discrete Sturm-Liouville operator generated in an infinite interval is studied. The usual condition for the existence of scattering is assumed to hold on the negative half-line and that the boundary problem generates a self-adjoint operator with a bounded discrete spectrum. The operator is assumed to have a continuous spectrum occupies the interval (-2,2) and a complex plane is found to have a branch cut along this interval. The solution of the problem is found to be linearly independent and that the function either has a simple pole or is regular. The set of quantities is called the scattering data for the operator and the inverse scattering problem determines the coefficients from the scattering data. It is found that a set of quantities is scattering data for an operator with coefficients satisfying some assumptions.