Schrödinger operators with complex singular potentials

Abstract

We study one-dimensional Schrödinger operators $\mathrm{S}(q)$ on the space $L^2(\mathbb{R})$ with potentials $q$ being complex-valued generalized functions from the negative space $H_{unif}^{-1}(\mathbb{R})$. Particularly the class $H_{unif}^{-1}(\mathbb{R})$ contains periodic and almost periodic $H_{loc}^{-1}(\mathbb{R})$-functions. We establish an equivalence of the various definitions of the operators $\mathrm{S}(q)$, investigate their approximation by operators with smooth potentials from the space $L_{unif}^1(\mathbb{R})$ and prove that the spectrum of each operator $\mathrm{S}(q)$ lies within a certain parabola.

arXiv comment11 pages. arXiv admin note: text overlap with arXiv:1111.0809 by other authors

References in corpus (1)

• Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials

Cited by in corpus (1)

• Schrödinger operators with measure-valued potentials: semiboundedness and spectrum