Generalized reflection coefficients

TL;DR

This paper introduces generalized reflection coefficients for one-dimensional operators, establishing their semicontinuity and implications for the spectrum, including applications to shift maps and integrable hierarchies like Toda and KdV.

Contribution

It generalizes the concept of reflection coefficients and proves their semicontinuity, extending previous results and applying to new classes of dynamical systems and hierarchies.

Findings

Reflection coefficients are semicontinuous functions of the operator.

Semicontinuity implies a semicontinuity result for the absolutely continuous spectrum.

Application to shift maps and integrable hierarchies like Toda and KdV.

Abstract

I consider general reflection coefficients for arbitrary one-dimensional whole line differential or difference operators of order $2$. These reflection coefficients are semicontinuous functions of the operator: their absolute value can only go down when limits are taken. This implies a corresponding semicontinuity result for the absolutely continuous spectrum, which applies to a very large class of maps. In particular, we can consider shift maps (thus recovering and generalizing a result of Last-Simon) and flows of the Toda and KdV hierarchies (this is new). Finally, I evaluate an attempt at finding a similar general setup that gives the much stronger conclusion of reflectionless limit operators in more specialized situations.