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).