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COMMON PART

Project Number21-71-10001

Project titleInverse spectral problems for differential operators with distribution coefficients

Project LeadBondarenko Natalia

AffiliationSaratov State University of N. G. Chernyshevsky,

Implementation period | 07.2021 - 06.2024 |

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-109 - Substantial and functional analysis

Keywordsspectral theory, nonlinear inverse problems, differential equations with coefficients-distributions, uniqueness theorems, constructive procedures, necessary and sufficient conditions of solvability

PROJECT CONTENT

Annotation

The present project relates to the spectral theory of differential operators. The goal of the project is construction of the theory for solving inverse problems of spectral analysis for differential operators of arbitrary orders with distribution coefficients. Inverse problems consist in recovering operators from certain their spectral characteristics. Such problems play a fundamental role in various areas of mathematics and have many applications in mechanics, physics, electronics, geophysics, meteorology and other branches of natural sciences and engineering. By the present, a general theory has been constructed for solving inverse spectral problems for differential operators of arbitrary orders with integrable coefficients. It should be noted that the difficulty of the study of inverse problems increases dramatically for operators of order higher than two. At the same time, inverse problems for certain important classes of operators, because of their difficulty, are studied insufficiently or have not been studied at all. In particular, there belong differential operators of higher order with coefficients from classes of generalized function. By now, a number of Russian and foreign mathematicians made a thorough study of direct and inverse spectral problems for differential operators of second order with generalized coefficients, understood in the sense of distributions. In particular, the analogs of many results have been obtained that were known for the classical Sturm–Liouville operator. At the same time, for differential operators of order higher than two with distribution coefficients the inverse problems have not been studied yet. The main reason for this a long time was the lack of a proper definition of such operators as well as the lack of necessary results for the corresponding direct spectral problems. However, now in the literature the understanding has been appeared how to define differential operators of higher order with coefficients-distributions. In addition, for them fundamental solutions of the Birkhoff type have been obtained. This fact opens wide perspectives for the study of inverse spectral problems for such operators, and this project is devoted to it. The research of the project participants is closely connected with inverse spectral problems. In particular, they obtained a solution of inverse problems for second-order differential operators with coefficients-distributions on geometrical graphs. During project implementation it is planned to use and to essentially expand methods and approaches developed by its participants. The results that are planned to be obtained during the project implementation will be new and allow solving nonlinear inverse problems for important and difficult for investigation classes of differential operators of arbitrary orders with distribution coefficients. Expected results will have both fundamental and applied importance in natural sciences and engineering.

Expected results

In the present project, it is planned to construct a theory for solving nonlinear inverse spectral problems for differential operators of arbitrary orders with distribution coefficients. The main expected results are uniqueness theorems for solutions of the inverse problems, constructive procedures for solving them and also necessary and sufficient conditions for their solvability. During the project implementation the main attention will be paid to studying the operators of higher order. It should be noted that the difficulty of the study of inverse problems increases dramatically for operators of order higher than two. It is supposed to investigate both the regular case of operators on a finite interval and singular operators on the half-line with more complicated behavior of the spectrum. The planned results will be of fundamental importance in mathematics and applications in science and engineering. Currently in the world there are a significant number of scientists specializing in the study of inverse spectral problems for various classes of operators. The results that are expected to be obtained during the implementation of this project will comply with world standards and stay ahead of analogous foreign developments in this area of mathematics. For the promulgation of the obtained results it is planned to publish several series of research papers in leading Russian and foreign peer-reviewed mathematical journals as well as to present the project results at international scientific conferences and simposia.

REPORTS

Annotation of the results obtained in 2023

In the third year of work on the project, a study was carried out on the solvability and stability of inverse spectral problems for higher-order differential operators with distribution coefficients, as well as on a number of related issues. The following main results have been obtained:
1. For higher-order differential operators with distribution coefficients, the most fundamental and difficult issue in the theory of inverse spectral problems has been solved - the necessary and sufficient conditions for the spectral data have been obtained. A class of differential operators of arbitrary orders n ≥ 2 (both even and odd) with coefficients from certain Sobolev spaces and separated boundary conditions was considered. The chosen class includes, in particular, the Sturm-Liouville operators with potentials from the space $W_2^{-1}(0,1)$. The spectral data for the n-th order differential equation are the eigenvalues of (n-1) boundary value problems and the corresponding weight numbers. The eigenvalues of each problem are assumed to be simple, and the spectra of neighboring problems do not intersect. The obtained necessary and sufficient conditions for the existence of solution to the inverse problem contain asymptotical and structural properties of the spectral data, as well as the requirement for the main equation solvability. In addition, the local solvability and stability of the inverse spectral problem are proven.
2. For the inverse problem from p. 1, necessary and sufficient conditions for the existence of solution without the requirement for the main equation solvability have been obtained in the self-adjoint case. For the fourth-order differential operators, the found conditions have the simplest form and include only the asymptotic behavior of the spectral data and their simplest structural properties. The theorem on the solvability of the main equation of the inverse problem is proven for a first-order differential system of general form. Therefore, this theorem can be applied to differential equations with coefficients from various classes of generalized functions. This result is novel even for the case of regular coefficients.
3. For the fourth-order differential operator, the uniqueness of solution is proven for the inverse problem by the eigenvalues and two sequences of norming constants, posed in the studies of J.R. McLaughlin in the 1980s. This problem has been interpreted within the framework of the general approach to inverse spectral problems for higher-order differential operators, which consists in recovering the operator coefficients from the Weyl-Yurko matrix by method of spectral mappings.
4. A constructive description is given of all possible matrices from a certain class, associated with a given differential expression in terms of the regularization approach of K.A. Mirzoev and A.A. Shkalikov. We have studied the influence of the choice of an associated matrix on the main spectral characteristics used in the inverse problem theory - the Weyl-Yurko matrix, discrete spectral data (eigenvalues and weight matrices). For higher-order differential operators with distribution coefficients, we have proven new uniqueness theorems for inverse problems, independent of the choice of associated matrices.
5. For the first-order differential system with summable coefficients on the semi-axis, we have obtained solutions that have exponential asymptotics and the property of analyticity in the spectral parameter. The property of quadratic summability is established for the remainders from the asymptotics under some additional conditions. Non-fundamental sets of solutions with analyticity in points of the double sector are also obtained. All constructed sets of solutions depend on an additional parameter, which makes it possible to prove the analyticity in points arbitrarily close to zero. The listed results have applications in the theory of inverse spectral problems for higher-order differential equations with distribution coefficients.
6. Local solvability and stability of the inverse Sturm-Liouville problems with distribution potential and polynomials of the spectral parameter in the boundary condition are proven. The eigenvalues and generalized weight numbers, which are the coefficients of the Laurent series for the Weyl function, are used as spectral data. The results were obtained both for the case of a simple spectrum and for the general case with multiple eigenvalues. In addition to this, for the Sturm-Liouville problem with polynomials in the boundary condition, the concept of generalized Cauchy data has been introduced, and the local solvability and stability of the inverse problem by the generalized Cauchy data have been proven.
7. A uniqueness theorem for solution of an inverse spectral problem has been proven for a new class of differential operators - the Sturm-Liouville operators with distribution potential and alternating weight.

Publications

**1.** *Bondarenko N.P.* **Inverse spectral problem for the third-order differential equation** Results in Mathematics, Vol. 78, Article number 179 (year - 2023) https://doi.org/10.1007/s00025-023-01955-x

**2.** *Bondarenko N.P.* **Regularization and inverse spectral problems for differential operators with distribution coefficients** Mathematics, Vol. 11, no. 16, Article ID 3455. (year - 2023) https://doi.org/10.3390/math11163455

**3.** *Bondarenko N.P.* **Local solvability and stability of an inverse spectral problem for higher-order differential operators** Mathematics, Vol. 11, no. 18, Article ID 3818. (year - 2023) https://doi.org/10.3390/math11183818

**4.** *Bondarenko N.P.* **Necessary and sufficient conditions for solvability of an inverse problem for higher-order differential operators** Mathematics, Vol. 12, no. 1, Article ID 61 (year - 2024) https://doi.org/10.3390/math12010061

**5.** *Bondarenko N.P.* **Spectral data asymptotics for fourth-order boundary value problems** The Bulletin of Irkutsk State University. Series Mathematics, Vol. 47, 31-46. (year - 2024) https://doi.org/10.26516/1997-7670.2024.47.31

**6.** *Chitorkin E.E.* **Обратная задача Штурма-Лиувилля с сингулярным потенциалом и многочленами от спектрального параметра в краевых условиях** Математика. Механика, Вып. 25. С. 103-108. (year - 2023)

**7.** *Chitorkin E.E., Bondarenko N.P.* **Local solvability and stability for the inverse Sturm-Liouville problem with polynomials in the boundary conditions** Mathematical Methods in the Applied Sciences, Published online (year - 2024) https://doi.org/10.1002/mma.10050

**8.** *Chitorkin E.E., Bondarenko N.P.* **Solving the inverse Sturm-Liouville problem with singular potential and with polynomials in the boundary conditions** Analysis and Mathematical Physics, Vol. 13, Article number: 79. (year - 2023) https://doi.org/10.1007/s13324-023-00845-3

**9.** *Ignatiev M.Yu.* **О спектральных данных систем дифференциальных уравнений с особенностью** Современные проблемы теории функций и их приложения, Вып. 22. С. 103-106 (year - 2024)

**10.** *Korsakov I.O.* **Единственность решения обратной задачи Штурма-Лиувилля с сингулярным потенциалом и точкой поворота** Математика. Механика, Вып. 25. С. 41-46. (year - 2023)

**11.** *Kuznetsova M.A.* **On solutions of systems of differential equations on half-line with summable coefficients** Lobachevskii Journal of Mathematics, - (year - 2024)

**12.** *Kuznetsova M.A.* **О решениях систем дифференциальных уравнений с нелинейной зависимостью от спектрального параметра** Современные проблемы теории функций и их приложения, Вып. 22. С. 140-143. (year - 2024)

**13.** *Bondarenko N.P.* **Partial inverse Sturm-Liouville problems** Mathematics, Vol. 11, no. 10, Article ID 2408. (year - 2023) https://doi.org/10.3390/math11102408

**14.** *Bondarenko N.P.* **Обратная спектральная задача для уравнения четвертого порядка** Материалы международной научной конференции "Уфимская осенняя математическая школа", Т. 1. С. 9-10. (year - 2023)

**15.** *Bondarenko N.P.* **Обратные задачи для дифференциальных операторов с коэффициентами-распределениями** Современные методы теории краевых задач. Понтрягинские чтения - XXXIV, С. 76-78 (year - 2023)

**16.** *Chitorkin E.E.* **Обратная задача Штурма-Лиувилля со спектральным параметром в краевых условиях** Современные методы теории краевых задач. Понтрягинские чтения - XXXIV, С. 436-437 (year - 2023)

**17.** *Kuznetsova M.A.* **Решения Биркгофа систем дифференциальных уравнений на полуоси** Материалы международной научной конференции "Уфимская осенняя математическая школа", Т. 1. С. 32-33 (year - 2023)

**18.** *Bondarenko N.P.* **McLaughlin’s inverse problem for the fourth-order differential operator** ArXiv, 24 p. (year - 2023) https://doi.org/10.48550/arXiv.2312.15988

**19.** *Chitorkin E.E., Bondarenko N.P.* **Inverse Sturm-Liouville problem with polynomials in the boundary condition and multiple eigenvalues** ArXiv, 25 p. (year - 2024) https://doi.org/10.48550/arXiv.2402.06215

Annotation of the results obtained in 2021

In the first year of the project implementation, we studied the properties of the spectral characteristics and the uniqueness of inverse spectral problem solution for the differential operators with distribution coefficients and also a number of related issues. The following main results have been obtained:
1. New properties of the spectral characteristics and the uniqueness theorems for the inverse spectral problems are proved for the higher-order differential operators with distribution coefficients. Differential operators of arbitrary even and odd orders on a finite interval and on the half-line have been considered. For each case, we have constructed the Weyl matrix which generalizes the Weyl function for the classical Sturm-Liouville operator and the Weyl-Yurko matrix for the higher-order differential operators with integrable coefficients. Analytical, asymptotical, and structural properties of the Weyl matrix are studied. For the cases of a finite interval and of the half-line, the uniqueness of the recovery of the differential expression coefficients and of the boundary condition coefficients from the Weyl matrix is proved. In the finite interval case, a certain part of the boundary condition coefficients have to be known a priori. Furthermore, in the finite interval case, the uniqueness theorem for the inverse problem by several spectra is proved.
2. Two papers on the inverse scattering problem for the system of differential equations with regular singularity have been prepared and accepted for publication. A constructive solution of the inverse scattering problem is obtained in the case of integrable potential without additional requirements of smoothness. The inverse problem is reduced to a linear equation in a Hilbert space. The invertibility is proved for the operator from the main equation. This implies the unique solvability of the main equation. A reconstruction formula for the potential is derived. The characteristic properties are obtained for the transfer matrix, which is used as the scattering data.
3. An approach to the investigation of inverse spectral problems for functional-differential operators with involution is developed. We have proved that the coefficients of the second-order functional-differential equations with involution-reflection are uniquely specified by the five spectra of the boundary value problems generated by the certain regular sets of the boundary conditions.
4. The uniqueness theorem is proved and an algorithm for the constructive solution is developed for the inverse spectral problem for the second-order differential pencil with distribution potential. This pencil has the quadratic dependence on the spectral parameter in the equation and entire analytic functions of the spectral parameter in one of the boundary conditions. The inverse problem consists in the recovery of the differential pencil coefficients from a part of the spectrum. This inverse problem statement generalizes a wide class of the so-called partial inverse problems, which consist in the recovery of differential operators from their spectral characteristics under the assumption that the differential expression coefficients are known a priori on a part of the interval. The obtained results are applied to the partial Hochstadt-Lieberman-type inverse problem for the quadratic pencil.
Thus, the tasks planned for the first year of the project implementation are fully completed. All the planned results together with some additional results have been obtained. The results are published in 5 papers, 2 of them are published in journals indexed by WOS and Scopus. Additionally, two papers are accepted for publication. 8 reports based on the project results have been presented at international scientific conferences and seminars.
Project Internet-resources:
[1] Bondarenko N.P. Inverse spectral problems for arbitrary-order differential operators with distribution coefficients, Mathematics 9 (2021), no. 22, Article ID 2989, https://doi.org/10.3390/math9222989
[2] Bondarenko N.P. Linear differential operators with distribution coefficients of various singularity orders, https://arxiv.org/abs/2204.02052
[3] Bondarenko N.P. Inverse spectral problems for functional-differential operators with involution, Journal of Differential Equations 318 (2022), 169-186, https://doi.org/10.1016/j.jde.2022.02.027
[4] Kuznetsova M.A. On recovering quadratic pencils with singular coefficients and entire functions in the boundary conditions, https://arxiv.org/abs/2111.05831

Publications

**1.** *Bondarenko N.P.* **Inverse spectral problems for arbitrary-order differential operators with distribution coefficients** Mathematics, Vol. 9, no. 22, Article ID 2989 (year - 2021) https://doi.org/10.3390/math9222989

**2.** *Bondarenko N.P.* **Inverse spectral problems for functional-differential operators with involution** Journal of Differential Equations, Vol. 318, P. 169-186 (year - 2022) https://doi.org/10.1016/j.jde.2022.02.027

**3.** *Ignatiev M.Yu.* **Конструктивное решение обратной задачи рассеяния для систем дифференциальных уравнений с особенностью** Вестник Московского университета. Серия 1. Математика. Механика, № 2. С. 24-34. (year - 2022) https://doi.org/10.55959/MSU0579-9368-1-64-2-3

**4.** *Bondarenko N.P.* **Неполные обратные задачи для дифференциальных операторов на графах** Современные проблемы теории функций и их приложения: материалы 21-й международной Саратовской зимней школы, С. 54-59 (year - 2022)

**5.** *Ignatiev M.Yu.* **О задаче рассеяния для систем дифференциальных уравнений с особенностью** Современные проблемы теории функций и их приложения: материалы 21-й международной Саратовской зимней школы, С. 134-138 (year - 2022)

**6.** *Kuznetsova M.A.* **Восстановление квадратичного дифференциального пучка с целыми функциями в краевых условиях** Современные проблемы теории функций и их приложения: материалы 21-й международной Саратовской зимней школы, С. 166-170 (year - 2022)

Annotation of the results obtained in 2022

In the second year of the project implementation, we continued to investigate inverse spectral problems for higher-order differential operators with distribution coefficients and also a number of related issues. The following main results have been obtained:
1. A method for constructive solution of inverse spectral problems for higher-order differential operators with distribution coefficients on a finite interval has been developed. As spectral data, we use eigenvalues of several boundary value problems and weight matrices (residues of the Weyl-Yurko matrix with respect to its poles). The nonlinear inverse problem is reduced to a linear equation in the Banach space of infinite sequences. We have proved that the operator which participates in the main equation is compact and has a bounded inverse. An explicit form of the inverse operator is obtained. As a result, the unique solvability of the main equation by necessity is proved. We have obtained formulas for the reconstruction of the differential operator coefficients by the main equation solution in form of series. We have proved that these series converge and that the constructed coefficients belong to required functional classes. As a result, a constructive algorithm for solving the inverse problem is developed. Note that the derivation of the main equation and the investigation of its solvability have been implemented for a sufficiently wide class of the first-order differential systems, to which higher-order differential equations with coefficients of various generalized function classes as well as with integrable coefficients can be reduced. The derivation of the reconstruction formulas, on the contrary, requires a separate construction for various operator classes, since it is necessary to obtain the convergence of the series in the corresponding functional spaces.
2. Asymptotical formulas have been obtained for the eigenvalues and the weight numbers of higher-order differential operators with distribution coefficients and with separated boundary conditions. We have estimated the differences of the spectral data for two problems such that a part of their coefficients in the differential expressions and in the boundary conditions coincide. These results are used for the constructive solution of the inverse problem but also have a separate significance.
3. The necessary and sufficient conditions of the inverse spectral problem solvability are obtained for the third-order differential operators with distribution coefficients in the general non-self-adjoint case. Local solvability and stability are proved for the inverse problem. In the self-adjoint case, we have obtained simple sufficient conditions for the global solvability. These conditions consist only of asymptotics and simple structural properties of the spectral data.
4. A regularization approach is developed for functional-differential operators with distribution coefficients. Such operators are fundamentally different from differential operators by their non-locality. Alghough the issues of regularization and spectral analysis for differential operators were studied for over 20 years, there were no results in this direction for functional-differential operators. The constructed regularization opens up perspectives of studying direct and inverse spectral problems for a wide class of operators, which can have several terms with constant delays and with fixed arguments.
5. Basing on the developed regularization approach, a spectral theory is constructed for the functional-differential Sturm-Liouville-type operators with a constant delay and with the potential of some distribution class. Note that this type of operators have been proviously studied only in the case of regular potentials. The eigenvalue asymptotics are obtained. The uniqueness of the distribution potential recovery from two spectra is proved for the case when the delay is not less 2/5 of the interval length. A constructive solution method is developed for the inverse problem. The necessary and sufficient conditions of its solvability are obtained. The uniform stability of the inverse problem is proved.
6. The uniqueness of solution is proved for the Sturm-Liouville inverse problem with polynomial dependence on the spectral parameter in one of the boundary conditions and with arbitrary entire analytic functions in the other one. The obtained results are applied to the Hochstadt-Lieberman-type inverse problems, which consist in the recovery of the potential on the half-interval from the spectrum (a subspectrum), while the potential on the other half-interval is known a priori. We have proved the uniqueness theorems for this type of problems with polynomial dependence on the spectral parameter not only in the boundary conditions but also in the discontinuity conditions inside the interval.
7. The uniqueness theorem is proved and a constructive solution method is developed for the inverse Sturm-Liouville problem with complex-valued potential of distributional class and with polynomials of the spectral parameter in the boundary conditions. The method consists in the reduction of the inverse spectral problem to a linear equation in the space of bounded infinite sequences and in the reconstruction of the potential and of the boundary condition poynomials by the solution of the main equation. Note that the obtained reconstruction formulas are novel even for the case of regular potential. They guarantee that the constructed potential and polynomials belong to required functional classes. Therefore, these formulas will be convenient for the future investigation of existence and stability of the inverse problem solution.
8. The difference of the matrix Sturm-Liouville potentials on the half-line is estimated in the case when the corresponding finite scattering data are close to each other. In this case, the eigenvalues of the two problems can have different multiplicities. The uniform stability of the inverse scattering problem by finite data is proved under some additional conditions on the potential and on the boundary condition coefficient.
9. The inverse spectral theory is constructed for the Laplace operator on the star-shaped graph with a nonlocal integral matching condition in the internal vertex. The inverse problem consists in the recovery of the integral condition coefficients from the spectrum. The uniqueness of the inverse problem solution is proved. Two constructive methods for solution are developed. The specrum characterization is obtained. This type of problems is principally new. Previously, for differential operators with integral boundary conditions, only the question of recovering the differential expression coefficients was studied, while the boundary condition coefficients were supposed to be known. Note that the considered operator on the star-shaped graph with the integral matching condition is adjoint to the functional-differential operator with “frozen” (fixed) argument at the internal vertex of the graph. Therefore, the obtained results and the developed methods can be applied to functional-differential operators with “frozen” arguments on intervals and on graphs.
Thus, the tasks planned for the second year of the project implementation are fully completed. All the planned results together with some additional results have been obtained. The results are published in 13 studies, 8 of them are papers in journals indexed by WOS and Scopus. 13 reports based on the project results have been presented at international scientific conferences and seminars.
Mathematical formulations of the results are provided in the attached papers and in the following project Internet-resources:
[1] Bondarenko N.P. Inverse spectral problem for the third-order differential equation, https://arxiv.org/abs/2303.13124
[2] Buterin S. Functional-differential operators with singular coefficients, https://www.mathnet.ru/PresentFiles/36919/ButerinSA.pdf
[3] Chitorkin E.E., Bondarenko N.P. Solving the inverse Sturm-Liouville problem with singular potential and with polynomials in the boundary conditions, https://arxiv.org/abs/2305.01231

Publications

**1.** *Bondarenko N.P.* **Reconstruction of higher-order differential operators by their spectral data** Mathematics, Vol. 10, no. 20, Article ID 3882 (year - 2022) https://doi.org/10.3390/math10203882

**2.** *Bondarenko N.P.* **Inverse problem for a differential operator on a star-shaped graph with nonlocal matching condition** Boletin de la Sociedad Matematica Mexicana, Vol. 29, Article number: 2. (year - 2023) https://doi.org/10.1007/s40590-022-00476-x

**3.** *Bondarenko N.P.* **Обратные спектральные задачи для дифференциальных операторов с коэффициентами-распределениями** Материалы международной научной конференции "Уфимская осенняя математическая школа", Т. 1. С. 18-20 (year - 2022) https://doi.org/10.33184/mnkuomsh1t-2022-09-28.5

**4.** *Bondarenko N.P.* **Linear differential operators with distribution coefficients of various singularity orders** Mathematical Methods in the Applied Sciences, Vol. 46, no. 6, 6639-6659 (year - 2023) https://doi.org/10.1002/mma.8929

**5.** *Bondarenko N.P.* **Spectral data asymptotics for the higher-order differential operators with distribution coefficients** Journal of Mathematical Sciences, Published online, 1-22 (year - 2023) https://doi.org/10.1007/s10958-022-06118-x

**6.** *Bondarenko N.P., Chitorkin E.E.* **Inverse Sturm-Liouville problem with spectral parameter in the boundary conditions** Mathematics, Vol. 11, no. 5, Article ID 1138 (year - 2023) https://doi.org/10.3390/math11051138

**7.** *Ignatiev M.Yu.* **О данных рассеяния дифференциальных систем с особенностью** Математические заметки, Т. 111, № 6. С. 846-863 (year - 2022) https://doi.org/10.4213/mzm13429

**8.** *Kuznetsova M.A.* **On recovering quadratic pencils with singular coefficients and entire functions in the boundary conditions** Mathematical Methods in the Applied Sciences, Vol. 46, no. 5, 5086-5098 (year - 2023) https://doi.org/10.1002/mma.8819

**9.** *Xu X.-C., Bondarenko N.P.* **Stability of the inverse scattering problem for the self-adjoint matrix Schrödinger operator on the half line** Studies in Applied Mathematics, Vol. 149, no. 3, 815-838 (year - 2022) https://doi.org/10.1111/sapm.12522

**10.** *Bondarenko N.P.* **Обратные задачи для дифференциальных операторов высших порядков с коэффициентами-распределениями** Вторая конференция Математических центров России : сборник тезисов, С. 31-32 (year - 2022)

**11.** *Bondarenko N.P.* **Обратная спектральная задача для функционально-дифференциального оператора с инволюцией** Современные методы теории краевых задач : материалы международной конференции «Понтрягинские чтения — XXXIII», С. 52-53 (year - 2022)

**12.** *Ignatiev M.Yu.* **Об одном методе построения решений систем обыкновенных дифференциальных уравнений** Международная конференция по дифференциальным уравнениям и динамическим системам, C. 127-128 (year - 2022)

**13.** *Ignatiev M.Yu.* **О поведении данных рассеяния систем с особенностью при малых значениях спектрального параметра** Материалы международной научной конференции "Уфимская осенняя математическая школа", Т. 1. С. 32-33 (year - 2022) https://doi.org/10.33184/mnkuomsh1t-2022-09-28.11