Hill’s potentials in Hörmander spaces and their spectral gaps - Molyboga, V. M.; Mikhailets, V. A. (Methods of Functional Analysis and Topology, 2011)
The paper deals with the Hill-Schrödinger operators with singular periodic potentials in the space H^ω(T)H^-1(T). The authors exactly describe the classes of sequences being the lengths of spectral gaps of these operators. The functions ω may be nonmonotonic. The space H^ω(T) coincides with the Hörmander space H_2^ω(T) with the weight function ω(1+ξ^2 ) if ω is in the Avakumovich class OR.