Generalized Toda flows

Abstract

The classical hierarchy of Toda flows can be thought of as an action of the (abelian) group of polynomials on Jacobi matrices. We present a generalization of this to the larger groups of C 2 and entire functions, and in this second case, we also introduce associated cocycles and in fact give center stage to this object.

On a description of the Toda hierarchy using cocycle maps

“…In particular, [Rem17] and [OR18] demonstrate the usefulness of this perspective The paper [Rem17] works out a Toda-type flow for canonical systems ([dB68]). Canonical systems are a spectral problem that generalizes the Jacobi equation.…”

“…A Toda-type flow thus is more sensibly expressed as an action on the m-functions rather than in terms of a Laxtype equation on a matrix operator. As another example, [OR18] uses this cocycle map point of view to develop a generalization of the Toda flow where each flow corresponds to a C 2 or a C ∞ function rather than just a polynomial. Acknowledgements.…”

“…In particular, it is often useful to treat the Toda flow as an evolution on the Weyl-Titchmarsh m-functions, rather than as an evolution on the Jacobi operator itself. See [Rem15], [Rem17] and [OR18] for examples of this approach. Given a Toda flow, we thus write m ± (z, t) as the m-functions corresponding to t ⊙ J.…”

Toda maps, cocycles, and canonical systems

“…While we will not provide a derivation of the general formula (5.1) (see [27,Section 12.4] and [18,Theorem A.1] for this), we will indicate in Section 3 how (2.9) can be obtained in the framework proposed in this paper.…”

“…More precisely, it is a compatibility condition that will ensure that the individual cocycles form a joint cocycle when glued together. Please see also Section 3 and especially Theorem 3.1 of [18], where these remarks are made more precise.…”