One-dimensional Schr\"odinger operators with singular periodic potentials

We study the one-dimensional Schr\"odinger operators $$S(q)u:=-u"+q(x)u,\quad u\in \mathrm{Dom}\left(S(q)\right),$$ with $1$-periodic real-valued singular potentials $q(x)\in H_{\operatorname{per}}^{-1}(\mathbb{R},\mathbb{R})$ on the Hilbert space $L_{2}\left(\mathbb{R}\right)$. We show equivalence of five basic definitions of the operators $S(q)$ and prove that they are self-adjoint. A new proof of continuity of the spectrum of the operators $S(q)$ is found. Endpoints of spectrum gaps are precisely described.