Hill's potentials in Hörmander spaces and their spectral gaps (Q2896643)

The classical Marchenko-Ostrovskii theorem gives a precise description of the connection between smoothness of a periodic potential of the Schrödinger-Hill operator and properties of the sequence of lengths of spectral gaps. The case of singular potentials from \(H^{-s}, \;0<s<1\), was considered by \textit{V. Mikhailets} and \textit{V. Molyboga} [Methods Funct. Anal. Topol. 15, No. 1, 31--40 (2009; Zbl 1199.34453)]. For singular potentials from \(H^{-1}\) belonging to Hörmander spaces \(H^\omega\) with monotone weights satisfying some regularity conditions, the above correspondence was established by \textit{P. Djakov} and \textit{B. Mityagin} [Dyn. Partial Differ. Equ. 6, No. 2, 95--165 (2009; Zbl 1238.34151)].NEWLINENEWLINEIn the paper under review, the authors extend the latter result to more general weights, removing the monotonicity and regularity assumptions. A description they give for some classes of Hörmander spaces on a circle is of independent interest.