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

---

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

---

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

---