Indefinite Sturm-Liouville operators with periodic coefficients

Mathematics – Spectral Theory

Scientific paper

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35 pages

Scientific paper

We investigate the spectral properties of the maximal operator $A$ associated with a differential expression $\frac{1}{w(-\frac{d}{dx}(p\frac{d}{dx}) + q)}$, where the coefficients $w$, $p$ and $q$ are real-valued and $w$ 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 $w$ and $p$ which ensures that $\infty$ is not a spectral singularity of $A$.

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