Gap Labelling for Almost Periodic Sturm-Liouville Operators

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.

💡 Research Summary

The paper develops a rotation‑number theory for Sturm–Liouville operators whose coefficients p, q, and w are Bohr almost‑periodic functions, extending the classic Johnson–Moser construction for Schrödinger operators. Because a Sturm–Liouville operator involves three coefficient functions, mere almost‑periodicity of each does not guarantee almost‑periodicity of the whole operator. To overcome this, the authors introduce the cone AP⁺(ℝ,ℝ) of positive almost‑periodic functions and require p, w ∈ AP⁺ while q ∈ AP. Lemma 2.2 shows that every element of the hull of p or w stays uniformly bounded away from zero, which in turn implies that 1/p and 1/w are also almost‑periodic and share the same frequency module.

Section 2 reviews the basic theory of almost‑periodic functions: the frequency module M_f, Fourier exponents, and the hull E(f). Lemma 2.1 establishes that the subspace of almost‑periodic functions with a prescribed module is a Banach algebra and isomorphic to C(E(f)). Lemma 2.4 provides a mean‑index formula for the zero‑density of an almost‑periodic function, a tool later used to define the rotation number.

The joint hull E(v) of the triple v = (p,q,w) is defined in (2.2)–(2.4). Lemma 2.10 proves that E(v) is a compact abelian topological group equipped with a uniquely ergodic R‑action and Haar measure μ_{E(v)}. This dynamical framework allows the authors to replace spatial averages by ergodic averages over the hull.

The first main result, Theorem 1.1, defines the Prüfer angle θ_λ(x;v)=arg(p(x)ϕ′(x)+iϕ(x)) for any non‑trivial solution ϕ of τ_{p,q,w}ϕ=λϕ and shows that the limit

 ρ(λ,v)=lim_{x→∞}