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Reflectionless Dirac operators and canonical systems

Christian Remling, Jie Zeng

Abstract

We study canonical systems that are reflectionless on an open set. In this situation, the two half line $m$ functions are holomorphic continuations of each other and may thus be combined into a single holomorphic function. This idea was explored in [11], and we continue these investigations here. We focus on Dirac operators and especially their interplay with canonical systems, and we provide a more general and abstract framework.

Reflectionless Dirac operators and canonical systems

Abstract

We study canonical systems that are reflectionless on an open set. In this situation, the two half line functions are holomorphic continuations of each other and may thus be combined into a single holomorphic function. This idea was explored in [11], and we continue these investigations here. We focus on Dirac operators and especially their interplay with canonical systems, and we provide a more general and abstract framework.

Paper Structure

This paper contains 6 sections, 20 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Dirac operators and canonical systems
  3. The space $\mathcal{R}(U)$
  4. Estimates on reflectionless Dirac potentials
  5. Proof of Theorem \ref{['T1.2']}, Dirac version
  6. General canonical systems

Key Result

Lemma 1.1

Let $U\subseteq{\mathbb R}$ be a non-empty open set, and assume that $H\in\mathcal{R} (U)$. Then the function has a holomorphic continuation to $\Omega\equiv{\mathbb C}^+\cup U\cup{\mathbb C}^-$.

Theorems & Definitions (38)

  • Lemma 1.1
  • Lemma 1.2
  • proof
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Definition 2.1
  • Theorem 2.3
  • ...and 28 more