Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients
In this paper, we consider a class of matrix functions that contains regularization matrices of Mirzoev and Shkalikov for differential operators with distribution coefficients of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">&...
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MDPI AG
2023-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/16/3455 |
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| author | Natalia P. Bondarenko |
| author_facet | Natalia P. Bondarenko |
| author_sort | Natalia P. Bondarenko |
| collection | DOAJ |
| description | In this paper, we consider a class of matrix functions that contains regularization matrices of Mirzoev and Shkalikov for differential operators with distribution coefficients of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>. We show that every matrix function of this class is associated with some differential expression. Moreover, we construct the family of associated matrices for a fixed differential expression. Furthermore, our regularization results are applied to inverse spectral theory. We study a new type of inverse spectral problems, which consist of the recovery of distribution coefficients from the spectral data independently of the associated matrix. The uniqueness theorems are proved for the inverse problems by the Weyl–Yurko matrix and by the discrete spectral data. As examples, we consider the cases <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula> in more detail. |
| first_indexed | 2024-03-10T23:46:18Z |
| format | Article |
| id | doaj.art-2a01d97372ef450e83ca682519ca2cef |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T23:46:18Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-2a01d97372ef450e83ca682519ca2cef2023-11-19T02:02:14ZengMDPI AGMathematics2227-73902023-08-011116345510.3390/math11163455Regularization and Inverse Spectral Problems for Differential Operators with Distribution CoefficientsNatalia P. Bondarenko0Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, RussiaIn this paper, we consider a class of matrix functions that contains regularization matrices of Mirzoev and Shkalikov for differential operators with distribution coefficients of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>. We show that every matrix function of this class is associated with some differential expression. Moreover, we construct the family of associated matrices for a fixed differential expression. Furthermore, our regularization results are applied to inverse spectral theory. We study a new type of inverse spectral problems, which consist of the recovery of distribution coefficients from the spectral data independently of the associated matrix. The uniqueness theorems are proved for the inverse problems by the Weyl–Yurko matrix and by the discrete spectral data. As examples, we consider the cases <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula> in more detail.https://www.mdpi.com/2227-7390/11/16/3455higher-order differential operatorsdistribution coefficientsregularizationinverse spectral problemsWeyl–Yurko matrixuniqueness theorem |
| spellingShingle | Natalia P. Bondarenko Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients Mathematics higher-order differential operators distribution coefficients regularization inverse spectral problems Weyl–Yurko matrix uniqueness theorem |
| title | Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients |
| title_full | Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients |
| title_fullStr | Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients |
| title_full_unstemmed | Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients |
| title_short | Regularization and Inverse Spectral Problems for Differential Operators with Distribution Coefficients |
| title_sort | regularization and inverse spectral problems for differential operators with distribution coefficients |
| topic | higher-order differential operators distribution coefficients regularization inverse spectral problems Weyl–Yurko matrix uniqueness theorem |
| url | https://www.mdpi.com/2227-7390/11/16/3455 |
| work_keys_str_mv | AT nataliapbondarenko regularizationandinversespectralproblemsfordifferentialoperatorswithdistributioncoefficients |