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One-dimensional Schrödinger operators with singular periodic potentials

Abstract

We study the one-dimensional Schrödinger operators $$S(q)u:=-u''+q(x)u,\quad u\in \mathrm{Dom}\left(S(q) \right),$$ with $1$-periodic real-valued singular potentials $q(x)\in H_{\operatorname{per}}^{-1}(\mathbb{R},\mathbb{R})$ 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.

Article Information

 Title One-dimensional Schrödinger operators with singular periodic potentials Source Methods Funct. Anal. Topology, Vol. 14 (2008), no. 2, 184-200 MathSciNet MR2432767 Milestones Received 13/03/2008 Copyright The Author(s) 2008 (CC BY-SA)

Authors Information

Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivs'ka, Kyiv, 01601, Ukraine

Volodymyr Molyboga
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivs'ka, Kyiv, 01601, Ukraine

Citation Example

Vladimir Mikhailets and Volodymyr Molyboga, One-dimensional Schrödinger operators with singular periodic potentials, Methods Funct. Anal. Topology 14 (2008), no. 2, 184-200.

BibTex

@article {MFAT465,
AUTHOR = {Mikhailets, Vladimir and Molyboga, Volodymyr},
TITLE = {One-dimensional Schrödinger operators with singular periodic potentials},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {14},
YEAR = {2008},
NUMBER = {2},
PAGES = {184-200},
ISSN = {1029-3531},
MRNUMBER = {MR2432767},
URL = {http://mfat.imath.kiev.ua/article/?id=465},
}