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).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 521–544, April, 2008.
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Mokhon’ko, O.A. Nonisospectral flows on semiinfinite unitary block Jacobi matrices. Ukr Math J 60, 598–622 (2008). https://doi.org/10.1007/s11253-008-0075-y
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DOI: https://doi.org/10.1007/s11253-008-0075-y