Сингулярно несиметрично збурені самоспряжені оператори
| dc.contributor.advisor | Дудкін, Микола Євгенович | |
| dc.contributor.author | Вдовенко, Тетяна Іванівна | |
| dc.date.accessioned | 2019-06-03T07:59:01Z | |
| dc.date.available | 2019-06-03T07:59:01Z | |
| dc.date.issued | 2019 | |
| dc.description.abstracten | During the last half century the theory of singularly perturbed operators – selfadjoint operators, in particular, with perturbations on the sets of zero Lebegue measure and their abstract counterparts has been sufficiently developed and studied. Laplace operator perturbed by δ -function is a typical example of this theory. The construction and research of perturbed selfadjoint operators in abstract case are presented in the works, of many authors: S. Albeverio, Yu. Arlinskii, J. Brasche, V.O. Derkach, M.E. Dudkin, W. Karwowski, V.D. Koshmanenko, P. Kurasov, S.O. Kuzhel, M.M. Malamud, H. Neidhart, L.P. Nizhnik, A. Posilicano, B. Simon, S. Hassi, E.R. Tsekanovskii, Y. Shondin and others. The total number of investigations devoted to singularly perturbed operators which have several thousand small and large articles. A large number of papers are devoted to perturbation of selfadjoint differential operators by δ -function potentials supported on different sets. The main method of investigation of such problems is the theory of selfadjoint extensions of densely defined symmetric operators. The consideration of models, in which singularly perturbed selfadjoint operator, but not symmetrically, leds to relevant singular nonsymmetric perturbations. Among these models there are models of physical control theory – systems with delay (anticipation), models with nonlocal interactions, perturbations by nonsymmetric potentials, which are in the works of S.O. Kuzhel, L.P. Nizhnik, V.O. Zolotarev. The transferring and, in some sence, the generalizing of results of the singularly perturbed selfadjoint operators to the case of nonsymmetric one, is a dificalt problem, starting from the definition of the corresponding operator and the investigation new not selfadjoint operators. The difficulty is in the absens of any analogs of the theory of selfadjoint extentions of densely defined symmetric operator for the case of nonsymmetric extentions. The thesis is devoted to singularly perturbed selfadjoint operators with nonsymmetric rank one perturbation that is a skew projection. Such perturbations are a generalization of the well-studied class of selfadjoint perturbations, in a particular case, it is reduced to them. Such nonsymmetric perturbations in functional spaces corresponds to the problems of non-local interactions. The technique of such perturbations correspond to models with delay (or anticipation). The description of nonsymmetrical rank one perturbations operator by generalized sum in the scale of spaces constructed on unperturbed operator is given in this work. The perturbed operator is described here by the resolvent M. Krein type formula , i.e., that is formulated and proven for selfadjoint extensions of symmetric densely defined operator. In the case, which has been M. Krein type formula represented in this work, combines resolvents of the selfadjoint not perturbed operators and the resolvent some nonsymmetric operator. As defect vectors in the M. Krein formula we have two defect vectors of symmetric operators that appear as a restriction on the dense subset not perturbed and perturbed operator and not perturbed adjoint operator and operator that is adjoint to perturbed one. Using the explicit form resolvents of singularly nonsymmetrically rank one perturbed operator there is studied the point spectrum that appears by perturbation. This spectrum in compare with the classical selfadjoint case, can be a complex number with the corresponding eigenvector and the complex conjugate number by adjoint operator but with another eigenvector. Also there is solved the inverse problem, which is that by given complex numbers and two vectors there is constructed singularly perturbed operator and its adjoint with given correspondence eigenvectors and eigenvalues. For this it is formulated and proved M. Krein type formula in the opposite direction, namely there are formulated necessary and sufficient conditions for the sum of the resolvent of not perturbed selfadjoint operator and skew projector (with certain properties) so that the obtained operator is the resolvent of nonsymmetrically rank one perturbed operator. By the investigation of the point spectrum there is founded the analog of the dual pair of eigenvalues for nonsymmetrically perturbed operator that generalizes the case of symmetrically perturbed operator. In this work dual pair of singularly nonsymmetrically rank one perturbed operator appears by the adjoint operator, respectively, with two other eigenvectors, in compare with the adjoint perturbed operator. Singularly nonsymmetrically rank one perturbations of selfadjoint operators are represented by two classes: wider one – a parameterically perturbed and a smollest class – uniquely perturbed. Division into classes is analog of a consideration of symmetrically singularly perturbed operators of class H_(-1) and H_(-2). All results are formulated separately for each class of operators. All results of this work are illustrated by numerous examples in which the role of not perturbed operator playes the operator of multiplication by the independent variable in the appropriate space. As vectors that form the perturbation are chosen the appropriate degree of polynomials. Also, the main results of the work are used for the model: the Laplace operator nonsymmetrically perturbed by δ -functions, namely, Laplace operator with nonlocal interactions. The cases Laplace operator in R^1 and R^3 are considered. Corresponding resolvent of perturbed operators are found. The point spectrum of nonsymmetrically perturbed Laplace operator is calculated. | uk |
| dc.description.abstractru | Диссертация посвящена исследованию сингулярно возмущенных самоспряженных операторов с несимметричным возмущением ранга один, т. е. косым проектором. Рассмотрено новые класы сингулярно возмущенных операторов. Для сингулярно возмущенных не симметричным слагаемым операторов рассмотрены два класса операторов, а именно: класс H_(-2) – требующий дополнительной параметризации, и класс H_(-1) – не требующий. Приведено описание таких операторов в терминах резольвенты. Полученные описания позволили исследовать точечный спектр, который возникает у сингулярно несимметрично ранга один возмущенного оператора. Также, решена обратная задача, тоесть по заданному набору собственных значений и собственных векторов построен сингулярно несимметрично ранга один возмущенный оператор. Найден аналог дуальной пары собственных значений для сингулярно возмущенного самосопряженного оператора и построен сингулярно несимметрично ранга один возмущенный оператор с дуальной парой. Приведенны примеры несимметрично возмущенного оператора Лапласа δ-функциями и с оператором умножения на независимую переменную. | uk |
| dc.description.abstractuk | Дисертація присвячена дослідженню сингулярно збурених самоспряжених операторів із несиметричним збуренням рангу один, тобто косим проектором. Розглянуто нові класи сингулярно збурених операторів. Для сингулярно несиметрично збурених операторів розглянуті два класи операторів: клас H_(-1) – який не вимагає додаткової параметризації, та клас H_(-2) – який вимагає. Цей клас означено та наведено опис таких операторів в термінах резольвенти. Отриманий опис дозволив дослідити точковий спектр сингулярно несиметрично рангу один збуреного оператора, який з’являється при таких збуреннях. Також розв’язана обернена задача, тобто за заданим набором власних значень та власних векторів побудовано сингулярно несиметрично рангу один збурений оператор. Знайдено аналог дуальної пари власних значень для оператора, сингулярно збуреного симетричним доданком та побудовано збурений оператор із дуальною парою, та оператором множення на незалежну змінну. Досліджено оператор Лапласа несиметрично збурений δ-функціями на дійсній вісі. | uk |
| dc.format.page | 134 с. | uk |
| dc.identifier.citation | Вдовенко, Т. І. Сингулярно несиметрично збурені самоспряжені оператори : дис. … канд. фіз.-мат. наук : 01.01.01 – математичний аналіз / Вдовенко Тетяна Іванівна. – Київ, 2019. – 134 с. | uk |
| dc.identifier.uri | https://ela.kpi.ua/handle/123456789/27790 | |
| dc.language.iso | uk | uk |
| dc.publisher.place | Київ | uk |
| dc.subject | сингулярне збурення | uk |
| dc.subject | несиметричне сингулярне збурення | uk |
| dc.subject | власні значення | uk |
| dc.subject | точковий спектр | uk |
| dc.subject | резольвента | uk |
| dc.subject | оператор Лапласа | uk |
| dc.subject | singular perturbation | uk |
| dc.subject | nonsymmetric singular perturbation | uk |
| dc.subject | eigenvalues | uk |
| dc.subject | resolvent | uk |
| dc.subject | Laplace operator | uk |
| dc.subject | сингулярное возмущения | uk |
| dc.subject | несимметричное сингулярное возмущение | uk |
| dc.subject | точечный спектр | uk |
| dc.subject | собственные значения | uk |
| dc.subject | резольвента | uk |
| dc.subject | оператор Лапласа | uk |
| dc.subject.udc | 517.9 (043.3) | uk |
| dc.title | Сингулярно несиметрично збурені самоспряжені оператори | uk |
| dc.type | Thesis Doctoral | uk |
Файли
Контейнер файлів
1 - 1 з 1
Ескіз недоступний
- Назва:
- Vdovenko_diss.docx
- Розмір:
- 532.05 KB
- Формат:
- Microsoft Word XML
- Опис:
Ліцензійна угода
1 - 1 з 1
Ескіз недоступний
- Назва:
- license.txt
- Розмір:
- 9.06 KB
- Формат:
- Item-specific license agreed upon to submission
- Опис: