Mokhonko O. Spectral theory of block Jacobi matrices and its application to difference-differential lattices integration problem

Thesis is devoted to the adaptation of the Inverse Spectral Problem method to solving the Cauchy problem for generalized Lax equation. Three main cases are considered: when the unknown is bounded self-adjoint, unitary and bounded normal operator. These options correspond to three main types of difference-differential lattices generated by Lax equation in weak sense: classical one-dimensional lattices (e.g. Toda lattice), block two-dimensional lattices and non-Abelian lattices of growing dimensions. The solution itself is given by the entries of the block Jacobi matrix of multiplication operator parameterized with time.