Spectral gaps of the Hill-Schrödinger operators with distributional potentials

Abstract

The paper studies the Hill-Schrödinger operators with potentials in the space $H^\omega \subset H^{-1}\left(\mathbb{T}, \mathbb{R}\right)$. The main results completely describe the sequences that arise as lengths of spectral gaps of these operators. The space $H^\omega$ coincides with the H\"{o}rmander space $H^{\omega}_2\left(\mathbb{T}, \mathbb{R}\right)$ with the weight function $\omega(\sqrt{1+\xi^{2}})$ if $\omega$ belongs to Avakumovich's class $\mathrm{OR}$. In particular, if the functions $\omega$ are power, then these spaces coincide with the Sobolev spaces. The functions $\omega$ may be nonmonotonic.