Nonisospectral flows on semiinfinite unitary block Jacobi matrices - Mokhonko, A. A. (Ukrains’kyi Matematychnyi Zhurnal, 2008)
It is proved that if the spectrum and spectral measure of a unitary operator generated by a semiinfinite block Jacobi matrix J(t) vary appropriately, then the corresponding operator J(t) satisfies the generalized Lax equation J (t) = Φ(J(t), t) + [J(t), A(J(t), t)], where Φ(λ, t) is a polynomial in λ and λ with t-dependent coefficients and A(J(t), t) = Ω + I + 12 Ψ is a skew-symmetric matrix. The operator J(t) is analyzed in the space CC^2C^2.... It is mapped into the unitary operator of multiplication L(t) in the isomorphic space L^2(T, dρ), where T = z: |z| = 1. This fact enables one to construct an efficient algorithm for solving the block lattice of differential equations generated by the Lax equation. A procedure that allows one to solve the corresponding Cauchy problem by the Inverse-Spectral-Problem method is presented. The article contains examples of block difference-differential lattices and the corresponding flows that are analogues of the Toda and van Moerbeke lattices (from self-adjoint case on R) and some notes about applying this technique for Schur flow (unitary case on T and OPUC theory).