Gap Labelling for Almost Periodic Sturm-Liouville Operators

TL;DR

This paper extends gap labeling theory to almost periodic Sturm-Liouville operators by introducing a rotation number and proving the gap labeling theorem, including the almost periodicity of Green's functions.

Contribution

It introduces a rotation number framework for almost periodic Sturm-Liouville operators and proves the gap labeling theorem in this context.

Findings

Established a rotation number for almost periodic Sturm-Liouville operators

Proved the gap labeling theorem using rotation numbers

Demonstrated the almost periodicity of Green's functions

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.