Nonisospectral flows on semiinfinite unitary block Jacobi matrices

Abstract

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 $$ \begin{array}{*{20}c} \cdot \\ J \\ \end{array} (t) = \Phi (J(t),t) + [J(t),A(J(t),t)] $$ , where Φ(gl, t) is a polynomial in λ and $$ \bar \lambda $$ with t-dependent coefficients and $$ A(J(t),t) = \Omega + I + \frac{1} {2}\Psi $$ 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 $$ L^2 (\mathbb{T},d\rho ) $$ , where $$ \mathbb{T} = \{ z:|z| = 1\} $$ . 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 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 $$ \mathbb{T} $$ and the OPUC theory).