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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.

### Article Information

 Title Eigenvalues of Schrödinger operators near thresholds: two term approximation Source Methods Funct. Anal. Topology, Vol. 26 (2020), no. 1, 76-87 DOI 10.31392/MFAT-npu26_1.2020.06 MathSciNet MR4113583 Milestones Received 05/09/2019; Revised 31/10/2019 Copyright The 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

### 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},
MRNUMBER = {MR4113583},
DOI = {10.31392/MFAT-npu26_1.2020.06},
URL = {http://mfat.imath.kiev.ua/article/?id=1290},
}

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