Spectral gaps of the Hill--Schr\"odinger operators with distributional potentials

TL;DR

This paper characterizes the spectral gap lengths of Hill--Schr"odinger operators with distributional potentials in certain function spaces, extending understanding to nonmonotonic weights and general distributional potentials.

Contribution

It provides a complete description of spectral gap sequences for operators with potentials in generalized Sobolev and Hörmander spaces, including nonmonotonic weight functions.

Findings

Spectral gap lengths are fully characterized for potentials in $H^ ext{omega}$ spaces.

The space $H^ ext{omega}$ aligns with Hörmander spaces with specific weight functions.

Results include cases with nonmonotonic weight functions.

Abstract

The paper studies the Hill--Schr\"odinger 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 H\"ormander 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.