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Remling's Theorem for vector-valued discrete Schrodinger operators

Keshav Raj Acharya

Abstract

This paper extends Remling's Theorem to vector-valued discrete Schrodinger operators, showing that the ω limit points of the matrix potentials, under the shift map, are reflectionless on the absolutely continuous spectrum with full multiplicity.

Remling's Theorem for vector-valued discrete Schrodinger operators

Abstract

This paper extends Remling's Theorem to vector-valued discrete Schrodinger operators, showing that the ω limit points of the matrix potentials, under the shift map, are reflectionless on the absolutely continuous spectrum with full multiplicity.
Paper Structure (4 sections, 15 theorems, 131 equations)

This paper contains 4 sections, 15 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Titchmarsh-Weyl $m$ function
  3. Topologies on the Space of Potentials
  4. Main theorems and their proofs

Key Result

Lemma 2.1

Let $\mathbb{N}_0 = \mathbb{N} \cup \{0\}$. For $F(n, z), G(n, z) \in \ell^2(\mathbb{N}_0, \mathbb{C}^{d\times d} )$, the following identity holds:

Theorems & Definitions (32)

  • Lemma 2.1: Green's Identity
  • proof
  • Definition 2.1
  • Theorem 2.1
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Definition 3.1
  • Proposition 3.1
  • proof
  • ...and 22 more