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.

Full Text

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

Vladimir Mikhailets
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},
}