Indefinite Sturm-Liouville operators with periodic coefficients

Abstract

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 $\infty$ is not a spectral singularity of $A$.