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Indefinite Sturm-Liouville operators with periodic coefficients

By Friedrich Philipp

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 $A$.Comment: 39 page

Topics: Mathematics - Spectral Theory, 34B09, 34B24, 34L10, 47B50, 47E05
Year: 2012
OAI identifier: oai:arXiv.org:1104.2286
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