Discrete skew selfadjoint canonical systems and the isotropic Heisenberg magnet model

A discrete analog of a skew selfadjoint canonical (Zakharov-Shabat or AKNS) system with a pseudo-exponential potential is introduced. For the corresponding Weyl function the direct and inverse problem are solved explicitly in terms of three parameter matrices. As an application explicit solutions are obtained for the discrete integrable nonlinear equation corresponding to the isotropic Heisenberg magnet model. State space techniques from mathematical system theory play an important role in the proofs. 0 Introduction In this paper we shall treat a discrete analog of the well-known skew selfad-joint canonical (Dirac type, Zakharov-Shabat or AKNS) system: −iJ dY dx (x, z) = zY (x, z) + V (x)Y (x, z), J = I m 0 0 −I m , x ≥ 0. (0.1) Here z is a spectral variable, Y and V are 2m × 2m matrix functions on the half line, and V is skew selfadjoint, that is, V (x) * = JV (x) with V (x) * being the matrix adjoint of V (x). To obtain the discrete analog of (0.1) let U be the unique solution of the initial value problem dU dx (x) = −iU(x)JV (x), x ≥ 0, U(0) = I 2m .