Reflectionless canonical systems, I. Arov gauge and right limits

TL;DR

This paper develops a general theory of $j$-monotonic matrix families in spectral theory, using canonical systems in Arov gauge, offering new insights into reflectionless properties and inverse spectral problems.

Contribution

It introduces a gauge-independent framework for $j$-monotonic families, connecting canonical systems with spectral properties and inverse problems in a novel way.

Findings

Provides a continuum analogue of the Schur algorithm.

Restores Schur functions along boundary value flows.

Offers a gauge-independent perspective on reflectionless spectra.

Abstract

In spectral theory, $j$-monotonic families of $2\times 2$ matrix functions appear as transfer matrices of many one-dimensional operators. We present a general theory of such families, in the perspective of canonical systems in Arov gauge. This system resembles a continuum version of the Schur algorithm, and allows to restore an arbitrary Schur function along the flow of associated boundary values at infinity. In addition to results in Arov gauge, this provides a gauge-independent perspective on the Krein-de Branges formula and the reflectionless property of right limits on the absolutely continuous spectrum. This work has applications to inverse spectral problems which have better behavior with respect to a normalization at an internal point of the resolvent domain.