# Hill&#x27;s potentials in Hörmander spaces and their spectral gaps - Mathematics > Spectral Theory

Abstract: In the paper we study the behaviour of the lengths of spectral gaps$\{\gamma {q}n\} {n\in \mathbb{N}}$ in a continuous spectrum of theHill-Schr\-{o}dinger operators $$Squ=-u-+qxu,\quad x\in\mathbb{R},$$ with1-periodic real-valued distribution potentials$$qx=\sum {k\in \mathbb{Z}}\hat{q}k e^{i k 2\pi x}\inH^{-1}\mathbb{T},\quad\text{and}\quad\hat{q}k=\bar{\hat{q}-k}, k\in\mathbb{Z},$$ in dependence on the weight $\omega$ of the H\-ormander spaces$H^{\omega}\mathbb{T} i q$, $\mathbb{T}=\mathbb{R}-\mathbb{Z}$.Let $h^{\omega}\mathbb{N}$ be a Hilbert space of weighted sequences. It isproved that $$\{\hat{q}\cdot\}\inh^{\omega}\mathbb{N}\Leftrightarrow\{\gamma {q}\cdot\}\inh^{\omega}\mathbb{N} \leqno\ast$$ if a positive, in general non-monotonic,weight $\omega=\{\omegak\} {k\in \mathbb{N}}$ is inter-power one.In the case $q\in L^{2}\mathbb{T}$, and $\omegak=1+2k^{s}$, $s\in\mathbb{Z} {+}$, the statement $\ast$ is due to Marchenko and Ostrovskii1975.