Determination of differential pencils with spectral parameter dependent boundary conditions from interior spectral data

Second‐order differential pencils L(p,q,h₀,h₁,H₀,H₁) on a finite interval with spectral parameter dependent boundary conditions are considered. We prove the following: (i) a set of values of eigenfunctions at the mid‐point of the interval [0,π] and one full spectrum suffice to determine differential pencils L(p,q,h₀,h₁,H₀,H₁); and (ii) some information on eigenfunctions at some an internal point and parts of two spectra suffice to determine differential pencils L(p,q,h₀,h₁,H₀,H₁). Copyright © 2013 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.     

Trace formulae for differential pencils with spectral parameter dependent boundary conditions

For the eigenvalues λ_n of a differential operator, the series , generally speaking, diverges; however, it can be regularized by subtracting from λ_n the first terms of the asymptotic expansion, which interfere with the convergence of the series. The sum of such a regularized series is called the trace of Gelfand–Levitan type. A second‐order differential pencil on a finite interval with spectral parameter dependent boundary conditions is considered. We derive the regularized trace formulae of Gelfand–Levitan type for this operator. Copyright © 2013 John Wiley & Sons, Ltd.     

Recovering nonselfadjoint differential pencils with nonseparated boundary conditions

Nonselfadjoint second-order differential pencils on a finite interval with nonseparated boundary conditions are studied. We establish some important properties of spectral characteristics and investigate inverse problems of recovering the operator from its spectral data. For these inverse problems, we prove the corresponding uniqueness theorems and provide procedures for constructing their solutions.     

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.     

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.     

Inverse problems for Sturm–Liouville operators with interior discontinuities and boundary conditions dependent on the spectral parameter

In this paper, we discuss the inverse problem for Sturm–Liouville operators with arbitrary number of interior discontinuities and boundary conditions having fractional linear function of spectral parameter on the finite interval [0,1]. Using Weyl function techniques, we establish some uniqueness theorems for the Sturm–Liouville operator. Copyright © 2012 John Wiley & Sons, Ltd.     

Partial inverse nodal problems for differential pencils on a star-shaped graph

In this paper, the authors study partial inverse nodal problems for differential pencils on a star-shaped graph. We firstly show that the potential on each edge can be uniquely determined by twin-dense nodal subsets on some interior intervals under certain conditions. Without any nodal information on some potential on the fixed edge, we may add some spectral information to guarantee these uniqueness theorems. We still consider the case of arbitrary intervals having the internal vertex. In particular, we pose and solve a new partial inverse nodal problem for differential pencils on the star-shaped graph from the Weyl m-function and the theory concerning densities of zeros of entire functions.     

An inverse spectral problem for a nonsymmetric differential operator: Reconstruction of eigenvalue problem

In this paper, an algorithm is established to reconstruct an eigenvalue problem from the given data satisfying certain conditions. These conditions are proved to be not only necessary but also sufficient for the given data to coincide with the spectral characteristics corresponding to the reconstructed eigenvalue problem.     

Inverse nodal and inverse spectral problems for discontinuous boundary value problems

Inverse nodal and inverse spectral problems are studied for second-order differential operators on a finite interval with discontinuity conditions inside the interval. Uniqueness theorems are proved, and a constructive procedure for the solution is provided.     

Inverse spectral problems for Dirac operator with eigenvalue dependent boundary and jump conditions

We deal with the Dirac operator with eigenvalue dependent boundary and jump conditions. Properties of eigenvalues, eigenfunctions and the resolvent operator are studied. Moreover, uniqueness theorems of the inverse problem according to the Weyl functions and the spectral data (the sets of eigenvalues and norming constants; two different eigenvalues sets) are proved.     

Inverse scattering problem for Sturm–Liouville operator with nonlinear dependence on the spectral parameter in the boundary condition

The inverse problem of the scattering theory for Sturm–Liouville operator on the half line with boundary condition depending quadratic on the spectral parameter is considered. Scattering data are defined, some properties of the scattering data are examined, the main equation is obtained, solvability of the integral equation is proved and uniqueness of algorithm to the potential with given scattering data is studied. Copyright © 2010 John Wiley & Sons, Ltd.     

二维椭圆型偏微分方程的反势问题

将Radon变换及其反投影变换原理应用于二维椭圆型偏微分方程反势问题的求解，从另一个角度解决了小扰动情况下椭圆型偏微分方程的反势问题．     

Inverse problems for the boundary value problem with the interior nodal subsets

In this paper, inverse nodal problems for Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter were studied. The authors showed that some uniqueness theorems on the potential function hold by the Weyl function, respectively.     

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.     

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.     

Inverse scattering with fixed energy and an inverse eigenvalue problem on the half-line

Recently A. G. Ramm (1999) has shown that a subset of phase shifts , , determines the potential if the indices of the known shifts satisfy the Müntz condition . We prove the necessity of this condition in some classes of potentials. The problem is reduced to an inverse eigenvalue problem for the half-line Schrödinger operators.

     

Inverse spectral problems for the Sturm-Liouville operator on a d-star graph

In this work, we consider inverse spectral problems for the Sturm-Liouville differential operator on a d-star-type graph with standard matching conditions in the internal vertex, where the integer d?2. By using the Yurko's method (Yurko (2008) [27], Yurko (2009) [28]) we show that

(1)

if the potential function qj(x) on a fixed edge ej is prescribed on the interval , then the reciprocal of d of the spectrum suffices to determine qj(x) on the whole interval [0,π];

(2)

the 2 over d of the spectrum suffices to determine qj(x) on a fixed edge ej.