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