Inverse Problem for Quasi-Periodic Differential Pencils with Jump Conditions Inside the Interval

Non-self-adjoint second-order differential pencils on a finite interval with non-separated quasi-periodic boundary conditions and jump conditions are studied. We establish properties of spectral characteristics and investigate the inverse spectral problem of recovering the operator from its spectral data. For this inverse problem we prove the corresponding uniqueness theorem and provide an algorithm for constructing its solution.     

The inverse problem for pencils of differential operators on the half-line with turning points

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning points is studied. We give a formulation of the inverse problem, establish properties of the spectral characteristics, and prove the uniqueness theorem for the solution of the inverse problem. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 237–246, 2006.     

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.     

Inverse spectral problems for differential pencils on the half-line with turning points

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning points is studied. We establish properties of the spectral characteristics, give a formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.     

Stability estimate for a multidimensional inverse spectral problem with partial spectral data

In Bellassoued, Choulli and Yamamoto (2009) [4] we proved a log-log type stability estimate for a multidimensional inverse spectral problem with partial spectral data for a Schrödinger operator, provided that the potential is known in a small neighbourhood of the boundary of the domain. In the present paper we discuss the same inverse problem. We show a log type stability estimate under an additional condition on potentials in terms of their X-ray transform. In proving our result, we follow the same method as in Alessandrini and Sylvester (1990) [1] and Bellassoued, Choulli and Yamamoto (2009) [4]. That is we relate the stability estimate for our inverse spectral problem to a stability estimate for an inverse problem consisting in the determination of the potential in a wave equation from a local Dirichlet to Neumann map (DN map in short).     

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.     

On inverse spectral problem for non-selfadjoint Sturm-Liouville operator on a finite interval

An inverse spectral problem is studied for a non-selfadjoint Sturm-Liouville operator on a finite interval with an arbitrary behavior of the spectrum. The spectral data introduced generalize the classical discrete spectral data corresponding to the specification of the spectral function in the selfadjoint case. The connection with other types of spectral characteristics is investigated and a uniqueness theorem is proved. A constructive procedure for solving the inverse problem is given.     

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 problems for Sturm–Liouville differential operators with a constant delay

We study non-self-adjoint second-order differential operators with a constant delay. We establish properties of the spectral characteristics and investigate the inverse problem of recovering operators from their spectra. The uniqueness theorem is proved for this inverse problem.     

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.     

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.     

Necessary and sufficient conditions for the solvability of the inverse problem for the matrix Sturm-Liouville operator

The matrix Sturm-Liouville operator on a finite interval with Dirichlet boundary conditions is studied. Properties of its spectral characteristics and the inverse problem of recovering the operator from these characteristics are investigated. Necessary and sufficient conditions on the spectral data of the operator are obtained. Research is conducted in the general case, with no a priori restrictions on the spectrum. A constructive algorithm for solving the inverse problem is provided.     

An inverse problem for Sturm-Liouville operators on arbitrary compact spatial networks

An inverse spectral problem is studied for Sturm-Liouville differential operators on arbitrary compact graphs (spatial networks). A uniqueness theorem of recovering operators from their spectra is proved, and a constructive procedure for the solution of the inverse problem is provided.     

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

The paper deals with the Sturm–Liouville eigenvalue problem with the Dirichlet boundary condition at one end of the interval and with the boundary condition containing entire functions of the spectral parameter at the other end. We study the inverse problem, which consists in recovering the potential from a part of the spectrum. This inverse problem generalizes partial inverse problems on finite intervals and on graphs and also the inverse transmission eigenvalue problem. We obtain sufficient conditions for global solvability of the studied inverse problem, which prove its local solvability and stability. In addition, application of our main results to the partial inverse Sturm–Liouville problem on the star-shaped graph is provided.     

Inverse Problems for Sturm–Liouville Operators on Noncompact Trees

We study Sturm–Liouville differential operators on noncompact graphs without cycles (i.e., on trees) with standard matching conditions in internal vertices. First we establish properties of the spectral characteristics and then we investigate the inverse problem of recovering the operator from the so-called Weyl vector. For this inverse problem we prove a uniqueness theorem and propose a procedure for constructing the solution using the method of spectral mappings. Received: February 13, 2007.     

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.     

An inverse problem for Sturm–Liouville differential operators with deviating argument

We consider second-order functional differential operators with a constant delay. Properties of their spectral characteristics are obtained and a nonlinear inverse problem is studied, which consists in recovering the operators from their spectra. We establish the uniqueness and develop a constructive algorithm for solution of the inverse problem.     

Inverse Scattering Theory for Discrete Schrödinger Operators on the Hexagonal Lattice

We consider the spectral theory and inverse scattering problem for discrete Schrödinger operators on the hexagonal lattice. We give a procedure for reconstructing finitely supported potentials from the scattering matrices for all energies. The same procedure is applicable for the inverse scattering problem on the triangle lattice.     

Reconstruction of an Impedance in Two-dimensions from Spectral Data

The two-dimensional spectral inverse problem involves the reconstruction of an unknown coefficient in an elliptic partial differential equation from spectral data, such as eigenvalues. Projection of the boundary value problem and the unknown coefficient onto appropriate vector spaces leads to a matrix inverse problem. Unique solutions of this matrix inverse problem exist provided that the eigenvalue data is close to the eigenvalues associated with the analogous constant coefficient boundary value problem. We discuss here the application of such a technique to the reconstruction of an impedance p in the boundary value problem $$ \eqalign{ -\nabla (\,p \nabla u) = \lambda p u \hbox {\quad in R} \cr u = 0 \hbox {\quad on R}}$$ where R is a rectangular domain. The matrix inverse problem, although nonstandard, is solved by a fixed-point iterative method and an impedance function p * is constructed which has the same m lowest eigenvalues as the unknown p . Numerical evidence of the success of the method will be presented.