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On self-adjontness of 1-D Schrödinger operators with $\delta$-interactions


Abstract

In the present work we consider the Schrödinger operator $\mathrm{H_{X,\alpha}}=-\mathrm{\frac{d^2}{dx^2}}+\sum_{n=1}^{\infty}\alpha_n\delta(x-x_n)$ acting in $L^2(\mathbb{R}_+)$. We investigate and complete the conditions of self-adjointness and nontriviality of deficiency indices for $\mathrm{H_{X,\alpha}}$ obtained in [13]. We generalize the conditions found earlier in the special case $d_n:=x_{n}-x_{n-1}=1/n$, $n\in \mathbb{N}$, to a wider class of sequences $\{x_n\}_{n=1}^\infty$. Namely, for $x_n=\frac{1}{n^{\gamma}\ln^\eta n}$ with $\langle\gamma,\eta \rangle\in(1/2,\,1)\!\times\!(-\infty,+\infty)\:\cup\:\{1\}\!\times\!(-\infty,1]$, the description of asymptotic behavior of the sequence $\{\alpha_n\}_{n=1}^{\infty}$ is obtained for $\mathrm{H_{X,\alpha}}$ either to be self-adjoint or to have nontrivial deficiency indices.


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Article Information

TitleOn self-adjontness of 1-D Schrödinger operators with $\delta$-interactions
SourceMethods Funct. Anal. Topology, Vol. 18 (2012), no. 4, 360-372
MathSciNet MR3058462
zbMATH 1289.34238
CopyrightThe Author(s) 2012 (CC BY-SA)

Authors Information

I. I. Karpenko
Tavrida National V. I. Vernadsky University, 4 Acad. Vernadsky Ave., Simferopol, 95007, Ukraine

D. L. Tyshkevich
Tavrida National V. I. Vernadsky University, 4 Acad. Vernadsky Ave., Simferopol, 95007, Ukraine 


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Citation Example

I. I. Karpenko and D. L. Tyshkevich, On self-adjontness of 1-D Schrödinger operators with $\delta$-interactions, Methods Funct. Anal. Topology 18 (2012), no. 4, 360-372.


BibTex

@article {MFAT623,
    AUTHOR = {Karpenko, I. I. and Tyshkevich, D. L.},
     TITLE = {On self-adjontness of 1-D Schrödinger operators with $\delta$-interactions},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {18},
      YEAR = {2012},
    NUMBER = {4},
     PAGES = {360-372},
      ISSN = {1029-3531},
  MRNUMBER = {MR3058462},
 ZBLNUMBER = {1289.34238},
       URL = {http://mfat.imath.kiev.ua/article/?id=623},
}


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