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The paper studies the Hill--Schr\"odinger operators with potentials in thespace $H^\omega \subset H^{-1}\left(\mathbb{T}, \mathbb{R}\right)$. The mainresults completely describe the sequences arising as the lengths of spectralgaps of these operators. The space $H^\omega$ coincides with the H\"ormanderspace $H^{\omega}_2\left(\mathbb{T}, \mathbb{R}\right)$ with the weightfunction $\omega(\sqrt{1+\xi^{2}})$ if $\omega$ belongs to Avakumovich's class$\mathrm{OR}$. In particular, if the functions $\omega$ are power, then thesespaces coincide with the Sobolev spaces. The functions $\omega$ may benonmonotonic.

Spectral gaps of the Hill--Schr\"odinger operators with distributional potentials