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.     

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 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.     

A Sturm-Liouville Inverse Spectral Problem with Boundary Conditions Depending on the Spectral Parameter

We consider a boundary value problem generated by the Sturm-Liouville equation on a finite interval. Both the equation and the boundary conditions depend quadratically on the spectral parameter. This boundary value problem occurs in the theory of small vibrations of a damped string. The inverse problem, i.e., the problem of recovering the equation and the boundary conditions from the given spectrum, is solved.     

On the Hochstadt–Lieberman Theorem for Discontinuous Boundary-valued Problems

In this paper, we discuss the half inverse problem for Sturm–Liouville equations with boundary conditions dependent on the spectral parameter and a finite number of discontinuities inside the interval and prove the Hochstadt–Liberman type theorem for the above boundary-valued problem.     

Identification of the polynomial in nonseparated boundary conditions by one eigenvalue

We consider a spectral problem for an ordinary differential equation on a finite interval. The boundary conditions contain functions and a polynomial in the spectral parameter. We find a criterion for the unique reconstruction of this polynomial by one multiple eigenvalue. Related examples are presented.     

A formula for the spectra of differential operators on graphs

We consider a connected undirected finite graph and a spectral problem generated by the double differentiation of functions on its edges (under usual conditions on the vertices ensuring the self-adjointness of the problem). We introduce, in a standard way, an entire function vanishing at the nonzero eigenvalues of the problem and give an explicit formula for this function, which involves graphs (introduced by V. I. Arnold) generated by a self-mapping of a finite set.     

Finite element approximation of the elasticity spectral problem on curved domains

We analyze the finite element approximation of the spectral problem for the linear elasticity equation with mixed boundary conditions on a curved non-convex domain. In the framework of the abstract spectral approximation theory, we obtain optimal order error estimates for the approximation of eigenvalues and eigenvectors. Two kinds of problems are considered: the discrete domain does not coincide with the real one and mixed boundary conditions are imposed. Some numerical results are presented.     

Nonlinear spectral problem for a self-adjoint vector differential equation

We consider a spectral problem that is nonlinear in the spectral parameter for a self-adjoint vector differential equation of order 2n. The boundary conditions depend on the spectral parameter and are self-adjoint as well. Under some conditions of monotonicity of the input data with respect to the spectral parameter, we present a method for counting the eigenvalues of the problem in a given interval. If the boundary conditions are independent of the spectral parameter, then we define the notion of number of an eigenvalue and give a method for computing this number as well as the set of numbers of all eigenvalues in a given interval. For an equation considered on an unbounded interval, under some additional assumptions, we present a method for approximating the original singular problem by a problem on a finite interval.     

Power moments of negative order for the principal spectral function of a string

We established necessary and sufficient conditions for the existence of finite power moments of all integer negative orders for the principal spectral function of a string. The necesity of this problem is explained by its relation to the so-called strong Stieltjes moment problem. Odessa Academy of Food Technologies, Odessa. Translated from Ukrainskii Matematischeskii Zhurnal, Vol. 48, No. 9, pp. 1209–1222, September, 1996.     

Inverse spectral problem of fourth‐order boundary value problems with finite spectrum

In the present paper, we consider a class of inverse spectral problem of fourth‐order boundary value problems. Under the so‐called “Atkinson type” conditions, the problem has finite spectrum and corresponding matrix representations. By using the method of inverse matrix eigenvalue problems of two‐banded matrix, the leading coefficient and potential functions are reconstructed from the given three sets of interlacing real numbers satisfying certain conditions.     

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.     

Asymptotic Behavior of Eigenvalues of a Boundary Value Problem for a Second-Order Elliptic Differential-Operator Equation with Spectral Parameter Quadratically Occurring in the Boundary Condition

The asymptotic behavior of eigenvalues of a boundary value problem for a secondorder differential-operator equation in a separable Hilbert space on a finite interval is studied for the case in which the same spectral parameter occurs linearly in the equation and quadratically in one of the boundary conditions. We prove that the problem has a sequence of eigenvalues converging to zero.     

Spectral computations: from operators to matrices

In this work we will address the problem of finding spectral values and bases for the corresponding spectral subspaces for a bounded operator on a Banach space. We will make a bridge between the spectral problem for the continuous operator and the computation of the eigenpairs of a matrix. This approximate problem results first from an approximation by a sequence of continuous operators with finite rank, followed by the reduction to a spectral problem for an operator whose domain as well as range are finite dimensional. Some discussion on defect correction procedures will be also presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Properties of eigenvalues and spectral singularities for impulsive quadratic pencil of difference operators

In this paper, we investigate the spectral analysis of impulsive quadratic pencil of difference operators. We first present a boundary value problem consisting one interior impulsive point on the whole axis corresponding to the above mentioned operator. After introducing the solutions of impulsive quadratic pencil of difference equation, we obtain the asymptotic equation of the function related to the Wronskian of these solutions to be helpful for further works, then we determine resolvent operator and continuous spectrum. Finally, we provide sufficient conditions guarenteeing finiteness of eigenvalues and spectral singularities by means of uniqueness theorems of analytic functions. The main aim of this paper is demonstrating the impulsive quadratic pencil of difference operator is of finite number of eigenvalues and spectral singularities with finite multiplicities which is an uninvestigated problem proposed in the literature.     

Schrödinger operators on graphs and geometry I: Essentially bounded potentials

The inverse spectral problem for Schrödinger operators on finite compact metric graphs is investigated. The relations between the spectral asymptotics and geometric properties of the underlying graph are studied. It is proven that the Euler characteristic of the graph can be calculated from the spectrum of the Schrödinger operator in the case of essentially bounded real potentials and standard boundary conditions at the vertices. Several generalizations of the presented results are discussed.     

Oscillation theorems for symplectic difference systems

We consider symplectic difference systems involving a spectral parameter, together with the Dirichlet boundary conditions. The main result of the paper is a discrete version of the so-called oscillation theorem which relates the number of finite eigenvalues less than a given number to the number of focal points of the principal solution of the symplectic system. In two recent papers the same problem was treated and an essential ingredient was to establish the concept of the multiplicity of a focal point. But there was still a rather restrictive condition needed, which is eliminated here by using the concept of finite eigenvalues (or zeros) from the theory of matrix pencils.     

Inverse spectral problem for the operators with non‐local potential

The main object under consideration in the paper is the second derivative operator on a finite interval with zero boundary conditions perturbed by a self‐adjoint integral operator with the degenerate kernel (non‐local potential). The inverse problem, i.e., the reconstruction of the perturbation from the spectral data, is solved by means of the step‐by‐step procedure based on the n‐interlacing property of the spectrum.     

On the isospectral problem of the dispersionless Camassa–Holm equation

We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa–Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa–Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.