Spectral gaps of the Hill-Schrödinger operators with distributional potentials - Molyboga, V. M.; Mikhailets, V. A. (Methods of Functional Analysis and Topology, 2014)
The paper studies the Hill-Schrödinger operators with potentials in the space H^ω H^-1(T, R). The main results completely describe the sequences that arise as lengths of spectral gaps of these operators. The space H^ω coincides with the Hörmander space H^ω_2(T, R) with the weight function ω(1+ξ^2 ) if ω belongs to Avakumovich’s class OR. In particular, if the functions ω are power, then these spaces coincide with the Sobolev spaces. The functions ω may be nonmonotonic.