Abstract
In this lecture we develop the method proposed in [1,2] for integrating nonlinear difference equations on semi-axis (n = 0,1,...) by means of the inverse spectral problem. In the first part of the paper (Subsections 1–3) we give the procedure for finding solutions to a wide class of difference equations connected with nonisospectral deformations of Jacobi matrices. This part is deeply connected with F.Calogero’s notion of spectral transformations (see [3]) and, more precisely, with the work [4]. In the second part of the paper (Subsection 4) we integrate the non-Abelian Toda chain on semi-axis in which the role of unknowns is played by a sequence of operator-valued functions of time. The system (16) from this part was proposed by A.Polyakov and was investigated on the axis (n =...,−1,0,1,...) in [5] by means of scattering theory and in [6] for periodic case.
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