Isospectral flows on a class of finite-dimensional Jacobi matrices

TL;DR

This paper introduces a new isospectral flow on finite-dimensional Jacobi matrices that asymptotically block-diagonalizes them, providing a unique limit structure and extending earlier theoretical work.

Contribution

It presents a novel matrix-valued ODE for Jacobi matrices that guarantees asymptotic block-diagonalization with a unique limit, extending prior research by Kac and van Moerbeke.

Findings

Solutions converge asymptotically to a block-diagonal form.

The limit matrix has a unique structure with sorted super-diagonal entries.

The method extends earlier work on matrix flows by Kac and van Moerbeke.

Abstract

We present a new matrix-valued isospectral ordinary differential equation that asymptotically block-diagonalizes $n\times n$ zero-diagonal Jacobi matrices employed as its initial condition. This o.d.e.\ features a right-hand side with a nested commutator of matrices, and structurally resembles the double-bracket o.d.e.\ studied by R.W.\ Brockett in 1991. We prove that its solutions converge asymptotically, that the limit is block-diagonal, and above all, that the limit matrix is defined uniquely as follows: For $n$ even, a block-diagonal matrix containing $2\times 2$ blocks, such that the super-diagonal entries are sorted by strictly increasing absolute value. Furthermore, the off-diagonal entries in these $2\times 2$ blocks have the same sign as the respective entries in the matrix employed as initial condition. For $n$ odd, there is one additional $1\times 1$ block containing a zero that is the top left entry of the limit matrix. The results presented here extend some early work of Kac and van Moerbeke.