Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum

Remarks on the Generalized Reflectionless Schrödinger Potentials

“…Third, there is an Abel map a which sends divisors d to unitary homomorphisms α (characters) defined on the fundamental group E of the multiply-connected domain E = C\E. This Abel map is that of Sodin-Yuditskii ( [60,61]; also [18]). When E is chosen appropriately, it is surjective and continuous but not injective; however it admits a right inverse i which, although discontinuous, is a pointwise limit of a sequence of continuous functions.…”

“…We consider domains of the form E = C\E where E ⊂ R is a closed subset with appropriate properties. We outline the construction of the Abel map a from the set D E of real divisors of E onto the set J E of unimodular characters on the fundamental group E of E [60,61]. In Sect.…”

“…There is a fixed character δ ∈ J E such that, if one defines a translation flow on J E by α → α + δx, then a is a homomorphism of flows. Now, if E is a homogeneous set in the sense of Carleson [2], then a is actually an isomorphism of flows [61]. We will produce an example of a set E of Parreau-Widom type, for which a is not injective but admits a right inverse of the first Baire class.…”

“…For example, if E consists of a finite number of nondegenerate closed intervals, and if q is reflectionless, then q is of algebro-geometric type (because it follows from (7) that condition (2) is valid; see [61] and Proposition 2.2 below). Let us note that there exist potentials q ∈ R which are not reflectionless; this is because the measure σ in (6 a )- (6 b ) can be any Borel regular measure supported on [−1, 1].…”

“…LevitanSavin [40], Moser [47] and Egorova [12] have studied Bohr almost periodic potentials q for which the operator L has Cantor spectrum. Sodin-Yuditskii [61] consider potentials q for which L has homogeneous spectrum; their q's are Bohr almost periodic.…”