On recovering Sturm—Liouville operators on graphs

Sturm—Liouville differential operators on compact graphs are studied. We establish properties of the spectral characteristics and investigate three inverse problems of recovering the operator from the so-called Weyl functions, from discrete spectral data, and from a system of spectra. For these inverse problems, we prove uniqueness theorems and obtain procedures for constructing the solutions by the method of spectral mappings.     

Inverse problems for the matrix Sturm–Liouville equation with a Bessel-type singularity

The matrix Sturm–Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. We consider inverse problems by the Weyl matrix and by the spectral data for this equation. Constructive solutions, based on the method of spectral mappings, are obtained for these inverse problems.     

A Remark on Inverse Problems for Sturm�CLiouville Operators on Graphs

We compare some formulations of the inverse spectral problems for differential operators on graphs with the classical inverse problems on an interval.     

Inverse problems for differential operators on trees with general matching conditions

Sturm–Liouville differential operators on compact trees with general matching conditions in internal vertices are studied. We establish properties of the spectral characteristics and investigate three inverse problems of recovering the operator either from the so-called Weyl functions, or from discrete spectral data or from a system of spectra. For these inverse problems, we prove the corresponding uniqueness theorems and obtain procedures for constructing their solutions by the method of spectral mappings.     

An inverse spectral problem for complex Jacobi matrices

We introduce the concept of generalized spectral function for finite order complex Jacobi matrices and solve the inverse problem with respect to the generalized spectral function. The results obtained can be used for solving of initial-boundary value problems for finite nonlinear Toda lattices with the complex-valued initial conditions by means of the inverse spectral problem method.     

Isoenergy Spectral Problem for Multidimensional Difference Operators

In this paper, the direct and inverse isoenergy spectral problems are solved for a class of multidimensional periodic difference operators. It is proved that the inverse spectral problem is solvable in terms of theta functions of curves added to the spectral variety under compactification, and multidimensional analogs of the Veselov–Novikov relations are found.     

Recovering variable order differential operators on star-type graphs from spectra

We consider inverse spectral problems for ordinary differential operators on compact star-type graphs for the case in which the differential equations have different orders on different edges. We study inverse problems of recovering potentials from a system of spectra. We provide algorithms for constructing solutions of these inverse problems and prove their uniqueness.     

Inverse problem for differential pencils on a hedgehog graph

We study boundary value problems on a hedgehog graph for second-order ordinary differential equations with a nonlinear dependence on the spectral parameter. We establish properties of spectral characteristics and consider the inverse spectral problem of reconstructing the coefficients of a differential pencil on the basis of spectral data. For this inverse problem, we prove a uniqueness theorem and obtain a procedure for constructing its solution.     

Spectral Analysis for an Indefinite Singular Sturm-Liouville Problem

Direct and inverse problems of spectral analysis are studied for an indefinite singular boundary value problem coming from astrophysics. We establish properties of the spectrum, prove completeness and expansion theorems and investigate the inverse problem of recovering the differential equation from the given spectral characteristics.     

Uniqueness Theorems for Sturm–Liouville Operators with Boundary Conditions Polynomially Dependent on the Eigenparameter from Spectral Data

In this paper, we discuss the inverse problems for Sturm–Liouville operators with boundary conditions polynomially dependent on the spectral parameter. We establish some uniqueness theorems on the potential q(x) for the half inverse problem and the interior inverse problem from spectral data, respectively.     

Recovery of differential equations with nonlinear dependence on the spectral parameter

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line is studied. We give a formulation of the inverse problem, prove the uniqueness theorem and provided a procedure for constructing the solution of the inverse problem. We also establishe connections with inverse problems for partial differential equations.     

Recovering differential pencils on compact graphs

We study boundary value problems on compact graphs without circles (i.e. on trees) for second-order ordinary differential equations with nonlinear dependence on the spectral parameter. We establish properties of the spectral characteristics and investigate the inverse spectral problem of recovering the coefficients of the differential equation from the so-called Weyl vector which is a generalization of the Weyl function (m-function) for the classical Sturm-Liouville operator. For this inverse problem we prove the uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mappings.     

Inverse problems on star-type graphs: differential operators of different orders on different edges

We study inverse spectral problems for ordinary differential equations on compact star-type graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.     

Canonical systems and transfer matrix-functions

The generalised Backlund-Darboux transform and its modifications are applied to the inverse spectral problems and problems on similarity of Hamiltonians for canonical systems.

     

Spectral method for the inverse vertical electrical sounding problem in two-dimensional quasi-layered media

The spectral method is considered for solving the direct and inverse problems of vertical electrical sounding (VES) in two-dimensional quasi-layered media. The behavior of the anomalous potential spectrum is analyzed. An example is presented applying the spectral method to solve the inverse VES problem. Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 34–43, 1999.     

Inverse spectral problems for differential systems on a finite interval

Inverse spectral problems are studied for non-selfadjoint systems of ordinary differential equations on a finite interval. We establish properties of the spectral characteristics, and provide a procedure for constructing the solution of the inverse problem of recovering the coefficients of differential systems from the given spectral characteristics.     

Inverse problem for Sturm-Liouville operators on hedgehog-type graphs

We study the inverse spectral problem for Sturm-Liouville differential operators on hedgehog-type graphs with a cycle and with standard matching conditions at interior vertices. We prove a uniqueness theorem and obtain a constructive solution for this class of inverse problems.     

Rational interpolation and solitons

We consider the general construction scheme for second-order spectral problems, for which the semiclassical approximation is exact. We show that the inverse spectral problem in this case reduces to the classical interpolation problem for meromorphic functions.     

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.     

Inverse spectral problem for differential operator pencils on noncompact spatial networks

We study boundary value problems on noncompact cycle-free graphs (i.e., trees) for second-order ordinary differential equations with a nonlinear dependence on the spectral parameter. We establish properties of the spectrum and analyze the inverse problem of reconstructing the coefficients of a differential equation on the basis of the so-called Weyl functions. For this inverse problem, we prove a uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mapping.