One-dimensional Schrödinger operators with singular periodic potentials

Abstract

We study the one-dimensional Schrödinger operators $$S(q)u:=-u"+q(x)u,u\in \mathrm{Dom}\left(S(q)\right),$$ with $1$-periodic real-valued singular potentials $q(x)\in H_{\mathrm{per}}^{-1}(\mathbb{R},\mathbb{R})$ on the Hilbert space $L_2\left(\mathbb{R}\right)$. We show equivalence of five basic definitions of the operators $S(q)$ and prove that they are self-adjoint. A new proof of continuity of the spectrum of the operators $S(q)$ is found. Endpoints of spectrum gaps are precisely described.

Published in Methods of Functional Analysis and Topology (MFAT), available at http://mfat.imath.kiev.ua/article/?id=465

References in corpus (1)

• Singularly perturbed periodic and semiperiodic differential operators

Cited by in corpus (1)

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