Reflectionless operators and automorphic Herglotz functions

TL;DR

This paper explores the mathematical structure of reflectionless operators and their connection to automorphic Herglotz functions, revealing insights into their topological properties and relation to finite gap operators.

Contribution

It establishes a novel correspondence between reflectionless operators and automorphic Herglotz functions via universal covers, enhancing understanding of their topological and measure-theoretic properties.

Findings

Characterization of reflectionless operators as automorphic Herglotz functions.

Insights into the topological space of reflectionless operators.

Analysis of the position of finite gap operators within this space.

Abstract

I am interested in canonical systems and Dirac operators that are reflectionless on an open set. In this situation, the half line $m$ functions are holomorphic continuations of each other and may be combined into a single function. By passing to the universal cover of its domain, we then obtain a one-to-one correspondence of these operators with Herglotz functions that are automorphic with respect to the Fuchsian group of covering transformations. I investigate the properties of this formalism, with particular emphasis given to the measures that are automorphic in a corresponding sense. This will shed light on the reflectionless operators as a topological space, on their extreme points, and on how the heavily studied smaller space of finite gap operators sits inside the (much) larger space.