On coupling constant thresholds in one dimension
The threshold behaviour of negative eigenvalues for Schr\"{o}dinger operators of the type \[ H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda\alpha_\lambda V(\alpha_\lambda \cdot) \] is considered. The potentials $U$ and $V$ are real-valued bounded functions of compact support, $\lambda$ is a positive pa...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2021-03-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
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| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/4126 |
| _version_ | 1797205816361615360 |
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| author | Yu.D. Golovaty |
| author_facet | Yu.D. Golovaty |
| author_sort | Yu.D. Golovaty |
| collection | DOAJ |
| description | The threshold behaviour of negative eigenvalues for Schr\"{o}dinger operators of the type \[ H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda\alpha_\lambda V(\alpha_\lambda \cdot) \] is considered. The potentials $U$ and $V$ are real-valued bounded functions of compact support, $\lambda$ is a positive parameter, and positive sequence $\alpha_\lambda$ has a finite or infinite limit as $\lambda\to 0$. Under certain conditions on the potentials there exists a bound state of $H_\lambda$ which is absorbed at the bottom of the continuous spectrum. For several cases of the limiting behaviour of sequence $\alpha_\lambda$, asymptotic formulas for the bound states are proved and the first order terms are computed explicitly. |
| first_indexed | 2024-04-24T08:57:08Z |
| format | Article |
| id | doaj.art-07773f83a2314c6f806ea4e00a4f6157 |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:57:08Z |
| publishDate | 2021-03-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-07773f83a2314c6f806ea4e00a4f61572024-04-16T07:05:54ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102021-03-01131223810.15330/cmp.13.1.22-383603On coupling constant thresholds in one dimensionYu.D. Golovaty0https://orcid.org/0000-0002-1758-0115Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, UkraineThe threshold behaviour of negative eigenvalues for Schr\"{o}dinger operators of the type \[ H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda\alpha_\lambda V(\alpha_\lambda \cdot) \] is considered. The potentials $U$ and $V$ are real-valued bounded functions of compact support, $\lambda$ is a positive parameter, and positive sequence $\alpha_\lambda$ has a finite or infinite limit as $\lambda\to 0$. Under certain conditions on the potentials there exists a bound state of $H_\lambda$ which is absorbed at the bottom of the continuous spectrum. For several cases of the limiting behaviour of sequence $\alpha_\lambda$, asymptotic formulas for the bound states are proved and the first order terms are computed explicitly.https://journals.pnu.edu.ua/index.php/cmp/article/view/41261d schrödinger operatorcoupling constant thresholdnegative eigenvaluezero-energy resonancehalf-bound state$\delta'$-potentialpoint interaction |
| spellingShingle | Yu.D. Golovaty On coupling constant thresholds in one dimension Karpatsʹkì Matematičnì Publìkacìï 1d schrödinger operator coupling constant threshold negative eigenvalue zero-energy resonance half-bound state $\delta'$-potential point interaction |
| title | On coupling constant thresholds in one dimension |
| title_full | On coupling constant thresholds in one dimension |
| title_fullStr | On coupling constant thresholds in one dimension |
| title_full_unstemmed | On coupling constant thresholds in one dimension |
| title_short | On coupling constant thresholds in one dimension |
| title_sort | on coupling constant thresholds in one dimension |
| topic | 1d schrödinger operator coupling constant threshold negative eigenvalue zero-energy resonance half-bound state $\delta'$-potential point interaction |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/4126 |
| work_keys_str_mv | AT yudgolovaty oncouplingconstantthresholdsinonedimension |