Indefinite Sturm-Liouville operators with periodic coefficients

We investigate the spectral properties of the maximal operator $A$ associated with a differential expression $\frac{1}{w}(-\frac{d}{dx}(p\frac{d}{dx})+q)$ with real-valued periodic coefficients $w$, $p$ and $q$ where $w$ changes sign. It turns out that the non-real spectrum of $A$ is bounded, symmetric with respect to the real axis and consists of a finite number of analytic curves. The real spectrum is band-shaped and neither bounded from above nor from below. We characterize the finite spectral singularities of $A$ and prove that there is only a finite number of them. Finally, we provide a condition on the coefficients which ensures that $\mathrm{\infty }$ is not a spectral singularity of $A$.