On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property
This article delves into the spectral problem associated with a multiple differentiation operator that features an integral perturbation of boundary conditions of one specific type, namely, regular but not strengthened regular. The integral perturbation is characterized by the function <inline-fo...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-10-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/11/20/4364 |
| _version_ | 1797573083570110464 |
|---|---|
| author | Makhmud Sadybekov Nurlan Imanbaev |
| author_facet | Makhmud Sadybekov Nurlan Imanbaev |
| author_sort | Makhmud Sadybekov |
| collection | DOAJ |
| description | This article delves into the spectral problem associated with a multiple differentiation operator that features an integral perturbation of boundary conditions of one specific type, namely, regular but not strengthened regular. The integral perturbation is characterized by the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow></semantics></math></inline-formula>, which belongs to the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula>. The concept of problems involving integral perturbations of boundary conditions has been the subject of previous studies, and the spectral properties of such problems have been examined in various early papers. What sets the problem under consideration apart is that the system of eigenfunctions for the unperturbed problem (when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>≡</mo><mn>0</mn></mrow></semantics></math></inline-formula>) lacks the property of forming a basis. To address this, a characteristic determinant for the spectral problem has been constructed. It has been established that the set of functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow></semantics></math></inline-formula>, for which the system of eigenfunctions of the perturbed problem does not constitute an unconditional basis in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula>, is dense within the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula>. Furthermore, it has been demonstrated that the adjoint operator shares a similar structure. |
| first_indexed | 2024-03-10T21:04:38Z |
| format | Article |
| id | doaj.art-55984f84f7134eba9d4e972c5020c896 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T21:04:38Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-55984f84f7134eba9d4e972c5020c8962023-11-19T17:14:57ZengMDPI AGMathematics2227-73902023-10-011120436410.3390/math11204364On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis PropertyMakhmud Sadybekov0Nurlan Imanbaev1Institute of Mathematics and Mathematical Modeling, Almaty 050010, KazakhstanInstitute of Mathematics and Mathematical Modeling, Almaty 050010, KazakhstanThis article delves into the spectral problem associated with a multiple differentiation operator that features an integral perturbation of boundary conditions of one specific type, namely, regular but not strengthened regular. The integral perturbation is characterized by the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow></semantics></math></inline-formula>, which belongs to the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula>. The concept of problems involving integral perturbations of boundary conditions has been the subject of previous studies, and the spectral properties of such problems have been examined in various early papers. What sets the problem under consideration apart is that the system of eigenfunctions for the unperturbed problem (when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mfenced open="(" close=")"><mi>x</mi></mfenced><mo>≡</mo><mn>0</mn></mrow></semantics></math></inline-formula>) lacks the property of forming a basis. To address this, a characteristic determinant for the spectral problem has been constructed. It has been established that the set of functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mfenced open="(" close=")"><mi>x</mi></mfenced></mrow></semantics></math></inline-formula>, for which the system of eigenfunctions of the perturbed problem does not constitute an unconditional basis in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula>, is dense within the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula>. Furthermore, it has been demonstrated that the adjoint operator shares a similar structure.https://www.mdpi.com/2227-7390/11/20/4364second-order differential operatoreigenvaluesystem of root vectorsbasis propertyintegral perturbation of boundary conditionscharacteristic determinant |
| spellingShingle | Makhmud Sadybekov Nurlan Imanbaev On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property Mathematics second-order differential operator eigenvalue system of root vectors basis property integral perturbation of boundary conditions characteristic determinant |
| title | On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property |
| title_full | On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property |
| title_fullStr | On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property |
| title_full_unstemmed | On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property |
| title_short | On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property |
| title_sort | on system of root vectors of perturbed regular second order differential operator not possessing basis property |
| topic | second-order differential operator eigenvalue system of root vectors basis property integral perturbation of boundary conditions characteristic determinant |
| url | https://www.mdpi.com/2227-7390/11/20/4364 |
| work_keys_str_mv | AT makhmudsadybekov onsystemofrootvectorsofperturbedregularsecondorderdifferentialoperatornotpossessingbasisproperty AT nurlanimanbaev onsystemofrootvectorsofperturbedregularsecondorderdifferentialoperatornotpossessingbasisproperty |