Inverse Problem Of Spectral Analysis Research Articles

Direct and Inverse Problems of Spectral Analysis for Arbitrary-Order Differential Operators with Nonintegrable Regular Singularities

Direct and Inverse Problems of Spectral Analysis for Arbitrary-Order Differential Operators with Nonintegrable Regular Singularities

Единственность восстановления оператора Штурма-Лиувилля со спектральным параметром, квадратично входящим в граничное условие

The work is devoted to the study of the inverse problem for the Sturm-Liouville operator with a real square-integrable potential. The boundary conditions are non-separated. One of these boundary conditions includes a quadratic function of the spectral parameter. A uniqueness theorem is proved and an algorithm for solving the inverse problem is constructed. As spectral data, we use the spectrum of the considered boundary value problem, the constant term of the quadratic function of the spectral parameter included in the boundary condition, and some special sequence of signs. From these spectral data, the characteristic function of the boundary value problem is first reconstructed in the form of an infinite product and the parameters of the boundary conditions, and then the problem is reduced to the inverse problem of reconstructing the potential of the Sturm-Liouville operator from the spectra of two boundary value problems with separated boundary conditions. The results of the article can be used for solving various versions of inverse problems of spectral analysis for differential operators, as well as for integrating some nonlinear equations of mathematical physics.

Direct and Inverse Problems of Spectral Analysis for Arbitrary-Order Differential Operators with Nonintegrable Regular Singularities

A short review is presented of results on the spectral theory of arbitrary order ordinary differential operators with non-integrable regular singularities. We establish properties of spectral characteristics, prove theorems on completeness of root functions in the corresponding spaces, prove expansion and equiconvergence theorems, and provide a solution of the inverse spectral problem for this class of operators.

Numerical solution and stability of the inverse spectral problem for a convolution integro-differential operator

In recent years, there has been growing interest in nonlinear inverse problems of spectral analysis for integro-differential operators. However, in spite of permanently increasing number of works, there are still no numerical results in this direction. The first aim of this paper is to fill this gap by developing an effective numerical approach to this class of inverse problems. The second aim is to prove the stability theorem, which theoretically justifies our numerical method. As a model situation, we consider one important and illustrative class of integro-differential operators, while the presented method can be extended to more complicated inverse problems. Our approach is based on reducing an inverse problem to some nonlinear integral equation and involves approximation of its solution by entire functions of exponential type. Concrete results of the numerical simulation are provided and discussed.

On the numerical solution of an inverse spectral problem with a singular potential

This paper deals with the unique solvability of the inverse problem for second-order differential operators having a singular potential. It is shown that the coefficients of the differential operator can be determined from the spectral data. Then, we construct the theoretical bases and provide a numerical algorithm for solving the inverse problem of spectral analysis and finding the potential function. Finally, some numerical examples are considered to demonstrate the applicability of the algorithm.

The Use of the Inverse Problem of Spectral Analysis to Forecast Time Series

The paper proposes a new method to forecast time series. We assume that a time series is a sequence of eigenvalues of a discrete self-adjoint operator acting in a Hilbert space. In order to construct such an operator, we use the theory of solving inverse problems of spectral analysis. The paper gives a theoretical justification for the proposed method. An algorithm for solving the inverse problem is given. Also, we give an example of constructing a differential operator whose eigenvalues practically coincide with a given time series.

On the Inverse Problem for Differential Operators on a Finite Interval with Complex Weights

Inverse problems of spectral analysis for second-order differential operators on a finite interval with complex-valued weights and with an arbitrary number of discontinuity conditions for the solutions inside the interval are studied. Properties of the spectral characteristics are established, and uniqueness theorems for this class of inverse problems are proved.

Inverse Spectral Problems for Differential Pencils on Arbitrary Compact Graphs

Inverse problems of spectral analysis are studied for second-order differential pencils on arbitrary compact graphs. The uniqueness of recovering operators from their spectra is proved, and a constructive procedure for the solution of this class of inverse problems is provided.

An inverse problem for the non-self-adjoint matrix Sturm-Liouville operator

The inverse problem of spectral analysis for the non-self-adjoint matrix Sturm-Liouville operator on a finite interval is investigated. We study properties of the spectral characteristics for the considered operator, and provide necessary and sufficient conditions for the solvability of the inverse problem. Our approach is based on the constructive solution of the inverse problem by the method of spectral mappings. The characterization of the spectral data in the self-adjoint case is given as a corollary of the main result.

Inverse spectral problems for Sturm–Liouville operators with complex weights

We study inverse problems of spectral analysis for second-order differential operators on a finite interval with complex-valued weights and with discontinuity conditions inside the interval. Properties of spectral characteristics are established, and the uniqueness theorems are proved for this class of non-linear inverse problems. A number of examples are provided.

Dirac system with a singularity in an interior point

We study the non-selfadjoint Dirac system on the line having an non-integrable regular singularity in an interior point with additional matching conditions at the singular point. Special fundamental systems of solutions are constructed with prescribed analytic and asymptotic properties. Behavior of the corresponding Stokes multipliers is established. These fundamental systems of solutions will be used for studying direct and inverse problems of spectral analysis.

On the solvability of an inverse problem in the central symmetric case

An inverse problem of spectral analysis is studied for Sturm–Liouville differential operators with non-separated boundary conditions. For the central symmetric case, we prove the local solvability of the inverse problem of recovering the potential from one spectrum.

Solution of the inverse quasiperiodic problem for the Dirac system

We present a complete solution of the inverse problem of spectral analysis for the Dirac operator with quasiperiodic boundary conditions. We prove a uniqueness theorem for the solution of the inverse problem and obtain necessary and sufficient conditions for a sequence of real numbers to be the spectrum of a quasiperiodic Dirac problem.

Reconstruction of Sturm-Liouville differential operators on A-graphs

We study the inverse problem of spectral analysis for Sturm-Liouville operators on A-graphs. We obtain a constructive procedure for solving the inverse problem of reconstruction of coefficients of differential operators from spectra and prove the uniqueness of the solution.

Inverse problem of spectral analysis of conflict dynamical systems

For conflict dynamical systems, we study the problem of the existence and description of initial measures that converge to measures with given spectral distributions.

An inverse problem for Sturm–Liouville differential operators on A-graphs

An inverse problem of spectral analysis is studied for Sturm–Liouville differential operators on a A-graph with the standard matching conditions for internal vertices. The uniqueness theorem is proved, and a constructive solution for this class of inverse problems is obtained.

Recovering Sturm-Liouville operators from spectra on a graph with a cycle

An inverse problem of spectral analysis is studied for Sturm-Liouville differential operators on a graph with a cycle and with generalized matching conditions at the internal vertex. Theorems on the unique recovery of operators from a system of spectra are proved, and a constructive solution is obtained for this class of inverse problems.Bibliography: 26 titles.

Inverse spectral problems for Sturm–Liouville differential operators on a finite interval

A short review is given on inverse problems of spectral analysis for the Sturm–Liouville differential operator on a finite interval. Using this operator as a model we present the main ideas and methods in the inverse problem theory, and describe the main results for this class of inverse problems.

Numerical Methods for Solving Inverse Sturm–Liouville Problems

A short review on numerical methods for solving inverse problems of spectral analysis for Sturm–Liouville differential operators is presented; moreover, a new approach based on ideas of the method of spectral mappings is given. Results of numerical experiments are also presented.

The inverse spectral problem for pencils of differential operators

The inverse problem of spectral analysis for a quadratic pencil of Sturm-Liouville operators on a finite interval is considered. A uniqueness theorem is proved, a solution algorithm is presented, and sufficient conditions for the solubility of the inverse problem are obtained.Bibliography: 31 titles.