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Eigenvalues of Schrödinger operators near thresholds: two term approximation


Abstract

We consider one dimensional Schrödinger operators \begin{equation*} H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda V_\lambda \end{equation*} with nonlinear dependence on the parameter $\lambda$ and study the small $\lambda$ behavior of eigenvalues. Potentials $U$ and $V_\lambda$ are real-valued bounded functions of compact support. Under some assumptions on $U$ and $V_\lambda$, we prove the existence of a negative eigenvalue that is absorbed at the bottom of the continuous spectrum as $\lambda\to 0$. We also construct two-term asymptotic formulas for the threshold eigenvalues.

Key words: 1D Schrödinger operator, coupling constant threshold, negative eigenvalue, zero-energy resonance, half-bound state.


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Article Information

TitleEigenvalues of Schrödinger operators near thresholds: two term approximation
SourceMethods Funct. Anal. Topology, Vol. 26 (2020), no. 1, 76-87
MilestonesReceived 05/09/2019; Revised 31/10/2019
CopyrightThe Author(s) 2020 (CC BY-SA)

Authors Information

Yuriy Golovaty
Department of Mechanics and Mathematics, Ivan Franko National University of Lviv, 1 Universytetska str., Lviv, 79000, Ukraine


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Citation Example

Yuriy Golovaty, Eigenvalues of Schrödinger operators near thresholds: two term approximation, Methods Funct. Anal. Topology 26 (2020), no. 1, 76-87.


BibTex

@article {MFAT1290,
    AUTHOR = {Yuriy Golovaty},
     TITLE = {Eigenvalues of Schrödinger operators near thresholds: two term 
approximation},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {26},
      YEAR = {2020},
    NUMBER = {1},
     PAGES = {76-87},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=1290},
}


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