Inverse Spectral Problem for Band Operators and Their Sparsity Criterion In Terms of Inverse Problem Data

We consider certain classes of operators generated by infinite band matrices (called band operators). They can be applied to the integration via the Lax pair formalism of nonlinear dynamical systems (e.g., Volterra type lattices) by using the inverse problem theory, in particular, by the inverse spectral problem method. A key role in this method is played by the moments of the Weyl matrix of a given band operator, which are used for unique reconstruction of the latter. These band operators have a special sparse structure, namely, they contain only two nonzero diagonals. We find a description of their sparsity in terms of these moments.