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).
Categories: math.SP