Spectral gaps of the one-dimensional Schrödinger operators with singular periodic potentials

The behaviour of the lengths of spectral gaps $\{γ_{n}(q)\}_{n=1}^{\infty}$ of the Hill-Schrödinger operators S(q)u=-u''+q(x)u,\quad u\in \mathrm{Dom}(S(q)) with real-valued 1-periodic distributional potentials $q(x)\in H_{1{-}per}^{-1}(\mathbb{R})$ is studied. We show that they exhibit the same behaviour as the Fourier coefficients $\{\widehat{q}(n)\}_{n=-\infty}^{\infty}$ of the potentials $q(x)$ with respect to the weighted sequence spaces $h^{s,φ}$, $s>-1$, $φ\in \mathrm{SV}$. The case $q(x)\in L_{1{-}per}^{2}(\mathbb{R})$, $s\in \mathbb{Z}_{+}$, $φ\equiv 1$ corresponds to the Marchenko-Ostrovskii Theorem.