# обратные спектральные задачи

## On Recovering Integro-Differential Operators from the Weyl Function

We study inverse problems of spectral analysis for second order integro-differential operators, which are a perturbation of the Sturm–Liouville operator by the integral Volterra operator. We pay the main attention to the nonlinear inverse problem of recovering the potential from the given Weyl function provided that the kernel of the integral operator is known a priori. We obtain properties of the spectral characteristics and the Weyl function, provide an algorithm for constructing the solution of the inverse problem and establish the uniqueness of the solution.

## The Inverse Problem of Spectral Analysis for the Matrix Sturm – Liouville Equation

The inverse spectral problem is investigated for the matrix Sturm - Liouville equation on a finite interval. The article provides properties of spectral characteristics, a constructive procedure for the solution of the inverse problem along with necessary and sufficient conditions for its solvability has been obtained.

## Uniqueness of the Solution of the Inverse Problem for Differential Operators on Arbitrary Compact Graphs

An inverse spectral problem is studied for Sturm – Liouville operators on arbitrary compact graphs with standard matching conditions in internal vertices. A uniqueness theorem of recovering operator’s coefficients from spectra is proved.

## Recovering Differential Operators on a Bush-Type Graph

An inverse spectral problem is studied for Sturm–Liouvilleoperators on arbitrary graphs with a cycle. A constructive procedure for the solution is provided and the uniquenness is established.

## On Recovering Differential Pencils on a Bush-type Graph

We study the inverse problem of spectral analysis for differential pencils on a bush-type graph, which is an arbitrary compact graph with one cycle. We pay the main attention to the most important nonlinear inverse problem of recovering coefficients of differential equations provided that the structure of the graph is known a priori. We use the standard matching conditions in the interior vertices and Dirichlet and Neumann boundary conditions in the boundary vertices.

## On Inverse Periodic Problem for Differential Operators for Central Symmetric Potentials

An inverse spectral problem for Sturm–Liouville operators on a finite interval with periodic boundary conditions is studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We discuss the statement of the problem, provide an algorithm for its solution along with necessary and sufficient conditions for the solvability of this nonlinear inverse problem.

## Uniqueness of Solution of the Inverse Scattering Problem for Various Order Differential Equation on the Simplest Noncompact Graph with Cycle

An inverse scattering problem is studied for variable orders differential operators on simplest noncompact graph with cycle. A uniqueness theorem of recovering coefficients of operators from the scattering data is provided.

## Uniqueness of recovering arbitrary order differential operators on noncompact spatial networks

An inverse spectral problem is studied for arbitrary order differential operators on noncompact graphs. A uniqueness theorem of recovering potentials from the Weyl matrices is proved.