Hill's potentials in Hörmander spaces and their spectral gaps

Abstract

The paper deals with the Hill-Schrödinger operators with singular periodic potentials in the space $H^{\omega}(\mathbb{T})\subset H^{-1}(\mathbb{T})$. The authors exactly describe the classes of sequences being the lengths of spectral gaps of these operators. The functions $\omega$ may be nonmonotonic. The space $H^{\omega}(\mathbb{T})$ coincides with the Hörmander space $H_{2}^{\omega}(\mathbb{T})$ with the weight function $\omega(\sqrt{1+\xi^{2}})$ if $\omega$ is in the Avakumovich class $\mathrm{OR}$.