Toda maps, cocycles, and canonical systems

Abstract

I present a discussion of the hierarchy of Toda flows that gives center stage to the associated cocycles and the maps they induce on the m functions. In the second part, these ideas are then applied to canonical systems; an important feature of this discussion will be my proposal that the role of the shift on Jacobi matrices should now be taken over by the more general class of twisted shifts.

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.…”

“…We demonstrate carefully that this new definition is equivalent to the traditional one. This paper may be viewed as a comapanion result to [Rem17] which discusses at greater length this alternative perspective for the Toda flow and a generalized flow on canonical systems.…”

“…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.…”

“…By theorem 2.2 of [Rem17], we can find a T that is a cocyle, that is, it obeys for any s, t, ∈ R, T (s + t, J) = T (s, t ⊙ J)T (t, J).…”