On an inverse problem for finite-difference operators of second order

The Jacobi matrices with bounded elements whose spectrum of multiplicity 2 is separated from its simple spectrum and contains an interval of absolutely continuous spectrum are considered. A new type of spectral data, which are analogous for scattering data, is introduced for this matrix. An integral equation that allows us to reconstruct the matrix from this spectral data is obtained. We use this equation to solve the Cauchy problem for the Toda lattice with the initial data that are not stabilized. Introduction 1. The Cauchy problem for the equation of the oscillation of the doubly-infinite Toda lattice d 2 xk d t2 = exk+1−xk − exk−xk−1 , k ∈ Z, (0.1) xk(0) = vk, ẋk(0) = wk , (0.2) with stabilized initial data ∞ ∑