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Spectral gaps of the Hill-Schrödinger operators with distributional potentials


Abstract

The paper studies the Hill-Schrödinger operators with potentials in the space $H^\omega \subset H^{-1}\left(\mathbb{T}, \mathbb{R}\right)$. The main results completely describe the sequences that arise as lengths of spectral gaps of these operators. The space $H^\omega$ coincides with the H\"{o}rmander space $H^{\omega}_2\left(\mathbb{T}, \mathbb{R}\right)$ with the weight function $\omega(\sqrt{1+\xi^{2}})$ if $\omega$ belongs to Avakumovich's class $\mathrm{OR}$. In particular, if the functions $\omega$ are power, then these spaces coincide with the Sobolev spaces. The functions $\omega$ may be nonmonotonic.

Key words: Hill–Schrödinger operator, singular potential, spectral gap, Hormanderspace.


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Article Information

TitleSpectral gaps of the Hill-Schrödinger operators with distributional potentials
SourceMethods Funct. Anal. Topology, Vol. 20 (2014), no. 4, 321-327
MathSciNet MR3309670
zbMATH 1324.47080
MilestonesReceived 08/09/2014
CopyrightThe Author(s) 2014 (CC BY-SA)

Authors Information

Vladimir Mikhailets
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine; National Technical University of Ukraine "Kyiv Polytechnic Institute", 37 Peremogy ave., Kyiv, 03056, Ukraine

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


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Citation Example

Vladimir Mikhailets and Volodymyr Molyboga, Spectral gaps of the Hill-Schrödinger operators with distributional potentials, Methods Funct. Anal. Topology 20 (2014), no. 4, 321-327.


BibTex

@article {MFAT754,
    AUTHOR = {Mikhailets, Vladimir and Molyboga, Volodymyr},
     TITLE = {Spectral gaps of the Hill-Schrödinger operators with distributional potentials},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {20},
      YEAR = {2014},
    NUMBER = {4},
     PAGES = {321-327},
      ISSN = {1029-3531},
  MRNUMBER = {MR3309670},
 ZBLNUMBER = {1324.47080},
       URL = {http://mfat.imath.kiev.ua/article/?id=754},
}


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