Discrete scattering theory (Q1089826)

The author develops the discrete and inverse scattering theory corresponding to a certain eigenvalue problem of the form \[ \psi (n+1,z)=J(z)\psi (n,z)+J(z)Q_ 0(n)\psi (n,z)+Q_ 1(n)\psi (n+1,z) \] where \(J(z)=diag(z,z^{-1})\) and \(Q_ 0\), \(Q_ 1\) and the unknown \(\psi\) are also \(2\times 2\) matrix-valued functions. This is used to solve the initial value problem for a class of differential-difference equations. The results are similar to those obtained for the continuous case, using different methods, by \textit{R. Beals} and \textit{R. R. Coifman} [Commun. Pure Appl. Math. 37, 39-90 (1984; Zbl 0514.34021)].