Spectral gaps of the Hill--Schrödinger operators with distributional potentials

Abstract

The paper studies the Hill--Schrödinger operators with potentials in the space $H^ω\subset H^{-1}\left(\mathbb{T}, \mathbb{R}\right)$. The main results completely describe the sequences arising as the lengths of spectral gaps of these operators. The space $H^ω$ coincides with the Hörmander space $H^ω_2\left(\mathbb{T}, \mathbb{R}\right)$ with the weight function $ω(\sqrt{1+ξ^{2}})$ if $ω$ belongs to Avakumovich's class $\mathrm{OR}$. In particular, if the functions $ω$ are power, then these spaces coincide with the Sobolev spaces. The functions $ω$ may be nonmonotonic.