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 arising as the lengths of spectral gaps of these operators. The space $H^{\omega }$ coincides with the Hörmander space $H_2^{\omega }\left(\mathbb{T},\mathbb{R}\right)$ 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.

6 pages

References in corpus (1)

• Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials