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Schrödinger operators with $(\alpha\delta'+\beta \delta)$-like potentials: norm resolvent convergence and solvable models

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Abstract

For real functions $\Phi$ and $\Psi$ that are integrable and compactly supported, we prove the norm resolvent convergence, as $\varepsilon\to0$, of a family $S_\varepsilon$ of one-dimensional Schrödinger operators on the line of the form $$ S_\varepsilon= -\frac{d^2}{d x^2}+\alpha\varepsilon^{-2}\Phi(\varepsilon^{-1}x)+\beta\varepsilon^{-1}\Psi(\varepsilon^{-1}x). $$ The limit results are shape-dependent: without reference to the convergence of potentials in the sense of distributions the limit operator $S_0$ exists and strongly depends on the pair $(\Phi,\Psi)$. A class of nontrivial point interactions which are formally related the pseudo-Hamiltonian $-\frac{d^2}{dx^2}+\alpha\delta'(x)+\beta\delta(x)$ is singled out. The limit behavior, as $\varepsilon\to 0$, of the scattering data for such potentials is also described.


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Article Information

TitleSchrödinger operators with $(\alpha\delta'+\beta \delta)$-like potentials: norm resolvent convergence and solvable models
SourceMethods Funct. Anal. Topology, Vol. 18 (2012), no. 3, 243-255
MathSciNet MR3051794
zbMATH 1265.34320
CopyrightThe Author(s) 2012 (CC BY-SA)

Authors Information

Yuriy Golovaty
Department of Mechanics and Mathematics, Ivan Franko National University of L'viv, 1 Universytets'ka, L'viv, 79000, Ukraine 


Citation Example

Yuriy Golovaty, Schrödinger operators with $(\alpha\delta'+\beta \delta)$-like potentials: norm resolvent convergence and solvable models, Methods Funct. Anal. Topology 18 (2012), no. 3, 243-255.


BibTex

@article {MFAT633,
    AUTHOR = {Golovaty, Yuriy},
     TITLE = {Schrödinger operators with  $(\alpha\delta'+\beta \delta)$-like potentials: norm resolvent convergence and solvable models},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {18},
      YEAR = {2012},
    NUMBER = {3},
     PAGES = {243-255},
      ISSN = {1029-3531},
  MRNUMBER = {MR3051794},
 ZBLNUMBER = {1265.34320},
       URL = {http://mfat.imath.kiev.ua/article/?id=633},
}


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