Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gap Labelling for Almost Periodic Sturm-Liouville Operators

Gerald Teschl, Yifei Wang, Bing Xie, Zhe Zhou

TL;DR

The paper defines a rotation number for almost periodic Sturm–Liouville operators and proves a gap-labelling theorem that ties spectral gaps to the frequency module of the coefficient triple (1/p, q, w). It develops a hull-based framework, shows the almost periodicity of Green’s functions, and reduces the spectral problem to a skew-product dynamical system via the Prüfer angle. Key contributions include establishing the continuity and ergodic representations of the rotation number and proving that 2ρ(λ,\mathbf{v}) belongs to the frequency module within each gap. This provides a robust, dynamical-analytic mechanism to label spectral gaps in a richly structured almost periodic setting with potential applications to broader classes of SL-type operators.

Abstract

In this paper, we introduce a rotation number for almost periodic Sturm-Liouville operators in the spirit of Johnson and Moser. We then prove the gap labelling theorem in terms of rotation numbers for the operator in question. To do this, we rigorously prove the almost periodicity of Green's functions.

Gap Labelling for Almost Periodic Sturm-Liouville Operators

TL;DR

The paper defines a rotation number for almost periodic Sturm–Liouville operators and proves a gap-labelling theorem that ties spectral gaps to the frequency module of the coefficient triple (1/p, q, w). It develops a hull-based framework, shows the almost periodicity of Green’s functions, and reduces the spectral problem to a skew-product dynamical system via the Prüfer angle. Key contributions include establishing the continuity and ergodic representations of the rotation number and proving that 2ρ(λ,\mathbf{v}) belongs to the frequency module within each gap. This provides a robust, dynamical-analytic mechanism to label spectral gaps in a richly structured almost periodic setting with potential applications to broader classes of SL-type operators.

Abstract

In this paper, we introduce a rotation number for almost periodic Sturm-Liouville operators in the spirit of Johnson and Moser. We then prove the gap labelling theorem in terms of rotation numbers for the operator in question. To do this, we rigorously prove the almost periodicity of Green's functions.
Paper Structure (14 sections, 27 theorems, 132 equations, 2 figures)

This paper contains 14 sections, 27 theorems, 132 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The setting and main results
  4. Almost periodic functions
  5. Module
  6. Joint hull
  7. Almost periodic Sturm-Liouville operators
  8. Operators associated with an element of the hull $\mathrm{E}(\textbf{v})$
  9. Green's function
  10. Rotation number
  11. Prüfer angle
  12. Reduction to skew-products
  13. Rotation number
  14. Gap labelling

Key Result

Theorem 1.1

The limit exists and is independent of the choice of solutions. We call it the rotation number of (sp-sl), and denote it by $\rho\left(\lambda,{\bf{v}}\right)$.

Figures (2)

  • Figure 1: $p(x)=\frac{1}{\sin x + 2}$, $q(x)=2\cos x$, $w(x)=-\cos x + 2$
  • Figure 2: $p(x)=\frac{1}{\sin x + 2}$, $q(x)=2\cos (\sqrt{2}x)$, $w(x)=-\cos x + 2$

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 43 more