Nonisospectral flows on semiinfinite unitary block Jacobi matrices

It is proved that if the spectrum and the 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 , where Φ(gl, t) is a polynomial in λ and with t-dependent coefficients and is a skew-symmetric matrix. The operator J(t) is analyzed in the space ℂ ⊕ ℂ² ⊕ ℂ² ⊕ …. It is mapped into the unitary operator of multiplication L(t) in the isomorphic space , where . 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 analogs of the Toda and the van Moerbeke lattices (from the self-adjoint case on ℝ) and some notes about the application of this technique to the Schur flow (the unitary case on and the OPUC theory). Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 521–544, April, 2008.