Spectral gaps of the Hill--Schr\\"odinger operators with distributional potentials
Vladimir Mikhailets, Volodymyr Molyboga · arXiv (Cornell University) · 2015
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.