Nonisospectral flows on semiinfinite unitary block Jacobi matrices
Abstract
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 $\textbf{J}(t)$ satisfies the generalized
Lax equation $\dot{\textbf{J}}(t) = \Phi(\textbf{J}(t), t) + [\textbf{J}(t), A(\textbf{J}(t), t)]$,
where $\Phi(\lambda, t)$ is a polynomial in $\lambda$ and $\overline{\lambda}$ with $t$-dependent coefficients and $A(J(t), t) = \Omega + I + \frac12 \Psi$ is a skew-symmetric matrix.
The operator $J(t$) is analyzed in the space ${\mathbb C}\oplus{\mathbb C}^2\oplus{\mathbb C}^2\oplus...$.
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 analogues of the Toda and van Moerbeke lattices
(from self-adjoint case on ${\mathbb R}$)
and some notes about applying this technique for Schur flow (unitary case on ${\mathbb T}$ and OPUC theory).
English version (Springer): Ukrainian Mathematical Journal 60 (2008), no. 4, pp 598-622.
Citation Example: Mokhonko A. A. Nonisospectral flows on semiinfinite unitary block Jacobi matrices // Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 521–544.
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