Scattering matrices for perturbations of Laplace operator by infinite sums of zero-range potentials
UDC 517.983, 517.984 We study the resolvents and scattering matrices for the following pairs of unbounded self-adjoint operators in $\mathbb{L}_{2}(\mathbf{R}_{3})$: the standard Laplace operator $A$ and its self-adjoint perturbation, which rigorously realizes the formal sum of $A$ with linear combi...
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| Date: | 2026 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/9170 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1871646540728107008 |
|---|---|
| author | Adamyan, V. Адамян, Вадим |
| author_facet | Adamyan, V. Адамян, Вадим |
| author_institution_txt_mv | [
{
"author": "V. Adamyan",
"institution": "Odesa Mechnikov National University"
}
] |
| author_sort | Adamyan, V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-07-24T15:24:13Z |
| description | UDC 517.983, 517.984
We study the resolvents and scattering matrices for the following pairs of unbounded self-adjoint operators in $\mathbb{L}_{2}(\mathbf{R}_{3})$: the standard Laplace operator $A$ and its self-adjoint perturbation, which rigorously realizes the formal sum of $A$ with linear combinations of infinitely many zero-range potentials. By using Krein’s resolvent formula for the perturbed operator, we establish sufficient conditions for the resolvent difference of the analyzed pair of operators to be a nuclear operator. Under these conditions, we deduce and analyze an explicit formula for the corresponding scattering matrices. |
| doi_str_mv | 10.3842/umzh.v78i7-8.9170 |
| first_indexed | 2026-07-25T01:00:35Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-9170 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-07-25T01:00:35Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-91702026-07-24T15:24:13Z Scattering matrices for perturbations of Laplace operator by infinite sums of zero-range potentials Матриці розсіяння для збурень оператора Лапласа нескінченними сумами потенціалів нульового радіуса Adamyan, V. Адамян, Вадим unbounded operator, B-Weyl, Left and right B-Weyl operators, Left and right Drazin invertible operators, Left and right B-Weyl spectra, Left and right Drazin spectra. unbounded operators банахів та гільбертів простори, нормальний оператор, задача Коші, спектр оператора, ціла вектор-функція, коректність n 1 раз проінтегрованої задачі Коші UDC 517.983, 517.984 We study the resolvents and scattering matrices for the following pairs of unbounded self-adjoint operators in $\mathbb{L}_{2}(\mathbf{R}_{3})$: the standard Laplace operator $A$ and its self-adjoint perturbation, which rigorously realizes the formal sum of $A$ with linear combinations of infinitely many zero-range potentials. By using Krein’s resolvent formula for the perturbed operator, we establish sufficient conditions for the resolvent difference of the analyzed pair of operators to be a nuclear operator. Under these conditions, we deduce and analyze an explicit formula for the corresponding scattering matrices. УДК 517.983, 517.984 Розглянуто резольвенти та матриці розсіювання для таких пар необмежених самоспряжених операторів у $\mathbb{L}_{2}(\mathbf{R}_{3})$: стандартного оператора Лапласа $A$ та його самоспряженого збурення, яке строго реалізує формальну суму $A$ з нескінченною лінійною комбінацією потенціалів нульового радіуса. З використанням формули Крейна для резольвенти збуреного оператора встановлено достатні умови, за яких різниця резольвент цієї пари є ядерним оператором. За цих умов виведено та проаналізовано явну формулу для відповідних матриць розсіювання. Institute of Mathematics, NAS of Ukraine 2026-07-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/9170 10.3842/umzh.v78i7-8.9170 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 7-8 (2026); 491–512 Український математичний журнал; Том 78 № 7-8 (2026); 491–512 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/9170/10696 Copyright (c) 2026 Вадим Адамян |
| spellingShingle | Adamyan, V. Адамян, Вадим Scattering matrices for perturbations of Laplace operator by infinite sums of zero-range potentials |
| title | Scattering matrices for perturbations of Laplace operator by infinite sums of zero-range potentials |
| title_alt | Матриці розсіяння для збурень оператора Лапласа нескінченними сумами потенціалів нульового радіуса |
| title_full | Scattering matrices for perturbations of Laplace operator by infinite sums of zero-range potentials |
| title_fullStr | Scattering matrices for perturbations of Laplace operator by infinite sums of zero-range potentials |
| title_full_unstemmed | Scattering matrices for perturbations of Laplace operator by infinite sums of zero-range potentials |
| title_short | Scattering matrices for perturbations of Laplace operator by infinite sums of zero-range potentials |
| title_sort | scattering matrices for perturbations of laplace operator by infinite sums of zero-range potentials |
| topic_facet | unbounded operator B-Weyl Left and right B-Weyl operators Left and right Drazin invertible operators Left and right B-Weyl spectra Left and right Drazin spectra. unbounded operators банахів та гільбертів простори нормальний оператор задача Коші спектр оператора ціла вектор-функція коректність n 1 раз проінтегрованої задачі Коші |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9170 |
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