Indefinite Sturm-Liouville operators with periodic coefficients

TL;DR

This paper studies the spectral properties of indefinite Sturm-Liouville operators with periodic coefficients, revealing the structure of their spectra, singularities, and conditions preventing spectral singularities at infinity.

Contribution

It provides a detailed analysis of the spectral characteristics of indefinite Sturm-Liouville operators with periodic coefficients, including the nature of their spectra and spectral singularities.

Findings

Non-real spectrum is bounded and symmetric about the real axis.

Real spectrum forms unbounded band-shaped regions.

Finite number of spectral singularities, with conditions to exclude spectral singularities at infinity.

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 $A$.