Resolution of the Cauchy problem for the Toda lattice with non-stabilized initial data

This paper is the continuation of the work ”On an inverse problem for finite-difference operators of second order” ([1]). We consider the Cauchy problem for the Toda lattice in the case when the corresponding L-operator is a Jacobi matrix with bounded elements, whose spectrum of multiplicity 2 is separated from its simple spectrum and contains an interval of absolutely continuous spectrum. Using the integral equation of the inverse problem for this matrix, obtained in the previous work, we solve the Cauchy problem for the Toda lattice with non-stabilized initial data. Introduction 1. In paper [1] we explained the importance of the extension of classes of initial data for which the Cauchy problem for nonlinear evolutionary equations can be solved. Due to this reason the goal of many investigations is the search of new inverse problems for linear L-operators, which can be applied to solve the corresponding Cauchy problems with new and possibly wider classes of initial data. In this connection in paper [1] we considered the Cauchy problem for the equation of oscillation of the doubly-infinite Toda lattice d 2 xk d t2 = ek+1k − ekk−1 , k ∈ Z, (0.1) xk(0) = vk, ẋk(0) = wk . (0.2) The Jacobi matrix