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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 $H_{{\operatorname{unif}}}^{-1}(\mathbb{R})$. Particularly the class $H_{{\operatorname{unif}}}^{-1}(\mathbb{R})$ contains periodic and almost periodic $H_{{\operatorname{loc}}}^{-1}(\mathbb{R})$-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 $L_{{\operatorname{unif}}}^{1}(\mathbb{R})$ and prove that the spectrum of each operator $\mathrm{S}(q)$ lies within a certain parabola.

Key words: 1-D Schr¨odinger operator, complex potential, distributional potential, resolvent approximation, localization of spectrum.


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Article Information

TitleSchrödinger operators with complex singular potentials
SourceMethods Funct. Anal. Topology, Vol. 19 (2013), no. 1, 16-28
MathSciNet MR3088075
zbMATH 1289.47093
MilestonesReceived 22/11/2012
CopyrightThe Author(s) 2013 (CC BY-SA)

Authors Information

Vladimir Mikhailets
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, 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, Schrödinger operators with complex singular potentials, Methods Funct. Anal. Topology 19 (2013), no. 1, 16-28.


BibTex

@article {MFAT677,
    AUTHOR = {Mikhailets, Vladimir and Molyboga, Volodymyr},
     TITLE = {Schrödinger operators with complex singular potentials},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {19},
      YEAR = {2013},
    NUMBER = {1},
     PAGES = {16-28},
      ISSN = {1029-3531},
  MRNUMBER = {MR3088075},
 ZBLNUMBER = {1289.47093},
       URL = {http://mfat.imath.kiev.ua/article/?id=677},
}


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