Hill's potentials in H\"ormander spaces and their spectral gaps

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 the Hill-Schr\"{o}dinger operators $$S(q)u=-u"+q(x)u,\quad x\in\mathbb{R},$$ with 1-periodic real-valued distribution potentials $$q(x)=\sum_{k\in \mathbb{Z}}\hat{q}(k) e^{i k 2\pi x}\in H^{-1}(\mathbb{T}),\quad\text{and}\quad\hat{q}(k)=\bar{\hat{q}(-k)}, k\in \mathbb{Z},$$ in dependence on the weight $\omega$ of the H\"ormander spaces $H^{\omega}(\mathbb{T})\ni q$, $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Let $h^{\omega}(\mathbb{N})$ be a Hilbert space of weighted sequences. It is proved that $$ \{\hat{q}(\cdot)\}\in h^{\omega}(\mathbb{N})\Leftrightarrow\{\gamma_{q}(\cdot)\}\in h^{\omega}(\mathbb{N}) \leqno(\ast) $$ if a positive, in general non-monotonic, weight $\omega=\{\omega(k)\}_{k\in \mathbb{N}}$ is inter-power one. In the case $q\in L^{2}(\mathbb{T})$, and $\omega(k)=(1+2k)^{s}$, $s\in \mathbb{Z}_{+}$, the statement $(\ast)$ is due to Marchenko and Ostrovskii (1975).