On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank
We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space ℋ. The perturbed operators \(\tilde A\) are defined by the Krein resolvent formula \((\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z \) , Im z ≠ 0, where B z are finite-rank operators such that d...
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| Datum: | 2003 |
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| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2003
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4000 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510129966809088 |
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| author | Dudkin, M. Ye. Koshmanenko, V. D. Дудкін, М. Є. Кошманенко, В. Д. |
| author_facet | Dudkin, M. Ye. Koshmanenko, V. D. Дудкін, М. Є. Кошманенко, В. Д. |
| author_sort | Dudkin, M. Ye. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:18:03Z |
| description | We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space ℋ. The perturbed operators \(\tilde A\) are defined by the Krein resolvent formula \((\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z \) , Im z ≠ 0, where B z are finite-rank operators such that dom B z ∩ dom A = |0}. For an arbitrary system of orthonormal vectors \(\{ \psi _i \} _{i = 1}^{n < \infty } \) satisfying the condition span |ψ i } ∩ dom A = |0} and an arbitrary collection of real numbers \({\lambda}_i \in {\mathbb{R}}^1\) , we construct an operator \(\tilde A\) that solves the eigenvalue problem \(\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n\) . We prove the uniqueness of \(\tilde A\) under the condition that rank B z = n. |
| first_indexed | 2026-03-24T02:52:06Z |
| format | Article |
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| id | umjimathkievua-article-4000 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:52:06Z |
| publishDate | 2003 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/63/cd646b2179ef721ebfad4a3b5a6b0563.pdf |
| spelling | umjimathkievua-article-40002020-03-18T20:18:03Z On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank Про точковий спектр самоспряжених операторів, що виникає при сингулярних збуреннях скінченного рангу Dudkin, M. Ye. Koshmanenko, V. D. Дудкін, М. Є. Кошманенко, В. Д. We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space ℋ. The perturbed operators \(\tilde A\) are defined by the Krein resolvent formula \((\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z \) , Im z ≠ 0, where B z are finite-rank operators such that dom B z ∩ dom A = |0}. For an arbitrary system of orthonormal vectors \(\{ \psi _i \} _{i = 1}^{n < \infty } \) satisfying the condition span |ψ i } ∩ dom A = |0} and an arbitrary collection of real numbers \({\lambda}_i \in {\mathbb{R}}^1\) , we construct an operator \(\tilde A\) that solves the eigenvalue problem \(\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n\) . We prove the uniqueness of \(\tilde A\) under the condition that rank B z = n. Розглядаються чисто сингулярні збурення скінченного рангу самоспряжеиого оператора $A$ в гільбертовому просторі $ℋ$. Збурені оператори $\tilde A$ визначаються формулою Крейна для резольвент $(\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z$, $Im z ≠ 0$ де $B_z$—оператори скінченного рангу такі, що $B_z \bigcap \text{dom} A = |0\}$. Для довільної системи ортонормованих векторів $\{ \psi _i \} _{i = 1}^{n < \infty }$ з умовою span $|ψ_i\} \bigcap \text{dom} A = |0\}$ та довільного набору дійсних чисел ${\lambda}_i \in {\mathbb{R}}^1$ побудовано оператор $\tilde A$ , який розв'язує задачу на власні значення: $\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n$. Доведено единість $\tilde A$ при умові, що ранг $B_z = n$. Institute of Mathematics, NAS of Ukraine 2003-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4000 Ukrains’kyi Matematychnyi Zhurnal; Vol. 55 No. 9 (2003); 1269-1276 Український математичний журнал; Том 55 № 9 (2003); 1269-1276 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4000/4712 https://umj.imath.kiev.ua/index.php/umj/article/view/4000/4713 Copyright (c) 2003 Dudkin M. Ye.; Koshmanenko V. D. |
| spellingShingle | Dudkin, M. Ye. Koshmanenko, V. D. Дудкін, М. Є. Кошманенко, В. Д. On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank |
| title | On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank |
| title_alt | Про точковий спектр самоспряжених операторів, що
виникає при сингулярних збуреннях скінченного рангу |
| title_full | On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank |
| title_fullStr | On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank |
| title_full_unstemmed | On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank |
| title_short | On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank |
| title_sort | on the point spectrum of self-adjoint operators that appears under singular perturbations of finite rank |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4000 |
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