Hill's potentials in Hörmander spaces and their spectral gaps

In the paper we study the behaviour of the lengths of spectral gaps $\{γ_{q}(n)\}_{n\in \mathbb{N}}$ in a continuous spectrum of the Hill-Schrödinger operators $S(q)u=-u"+q(x)u,\quad x\in\mathbb{R},$ with 1-periodic real-valued distribution potentials $q(x)=\sum_{k\in \mathbb{Z}}\hat{q}(k) e^{i k 2πx}\in H^{-1}(\mathbb{T}),\quad\text{and}\quad\hat{q}(k)=\bar{\hat{q}(-k)}, k\in \mathbb{Z},$ in dependence on the weight $ω$ of the Hörmander spaces $H^ω(\mathbb{T})\ni q$, $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Let $h^ω(\mathbb{N})$ be a Hilbert space of weighted sequences. It is proved that $$ \{\hat{q}(\cdot)\}\in h^ω(\mathbb{N})\Leftrightarrow\{γ_{q}(\cdot)\}\in h^ω(\mathbb{N}) \leqno(\ast) $$ if a positive, in general non-monotonic, weight $ω=\{ω(k)\}_{k\in \mathbb{N}}$ is inter-power one. In the case $q\in L^{2}(\mathbb{T})$, and $ω(k)=(1+2k)^{s}$, $s\in \mathbb{Z}_{+}$, the statement $(\ast)$ is due to Marchenko and Ostrovskii (1975).

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