Differential Pencils Research Articles

Inverse spectral problem for differential pencils with a frozen argument

Abstract This paper deals with differential pencils possessing a term depending on the unknown function with a fixed argument. We deduce the so called main equation together with its fine structure for the spectral problem. Then, according to the boundary conditions and the position of argument, we describe two cases: degenerate and non-degenerate. For these two cases, the uniqueness of inverse spectral problem is studied and a constructive procedure for reconstructing the potentials along with necessary and sufficient conditions of the inverse problem solvability are obtained.

Trace Formulae for Second-Order Differential Pencils with a Frozen Argument

This paper deals with second-order differential pencils with a fixed frozen argument on a finite interval. We obtain the trace formulae under four boundary conditions: Dirichlet–Dirichlet, Neumann–Neumann, Dirichlet–Neumann, Neumann–Dirichlet. Although the boundary conditions and the corresponding asymptotic behaviour of the eigenvalues are different, the trace formulae have the same form which reveals the impact of the frozen argument.

On recovering quadratic pencils with singular coefficients and entire functions in the boundary conditions

The paper deals with a new type of inverse spectral problems for second‐order quadratic differential pencils when one of the boundary conditions involves arbitrary entire functions of the spectral parameter. Although various aspects of the inverse spectral theory for the pencils have been of a special interest during the last decades, such settings were considered before only in the particular case of a Sturm–Liouville equation. We develop an approach covering also the quadratic dependence on the spectral parameter in the differential equation, which is based on the completeness and basisness of certain functional systems. By this approach, we obtain a uniqueness theorem and an algorithm for solving the inverse problem along with sufficient properties of the mentioned systems. The presented results give a universal tool for studying a number of important specific situations, including various Hochstadt–Lieberman‐type inverse problems both on an interval and on geometrical graphs, which is illustrated as well.

Recovering differential pencils with spectral boundary conditions and spectral jump conditions

In this work, we discuss the inverse problem for second order differential pencils with boundary and jump conditions dependent on the spectral parameter. We establish the following uniqueness theorems: (i) the potentials q_{k}(x) and boundary conditions of such a problem can be uniquely established by some information on eigenfunctions at some internal point bin (frac{pi }{2},pi ) and parts of two spectra; (ii) if one boundary condition and the potentials q_{k}(x) are prescribed on the interval [pi /2(1-alpha ),pi ] for some alpha in (0, 1), then parts of spectra Ssubseteq sigma (L) are enough to determine the potentials q_{k}(x) on the whole interval [0, pi ] and another boundary condition.

Inverse spectral problems for differential operators with non-separated boundary conditions

Abstract We give a short review of results on inverse spectral problems for second-order differential operators on an interval with non-separated boundary conditions. We pay the main attention to the most important nonlinear inverse problems of recovering coefficients of differential operators from given spectral characteristics. In the first part of the review, we provide the main results and methods related to inverse problems for Sturm–Liouville operators with non-separated boundary conditions: periodic, quasi-periodic and Robin-type boundary conditions. At the end, we present the main results on inverse problems for differential pencils with non-separated boundary conditions.

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.

Partial inverse problems for quadratic differential pencils on a graph with a loop

Abstract In this paper, partial inverse problems for the quadratic pencil of Sturm–Liouville operators on a graph with a loop are studied. These problems consist in recovering the pencil coefficients on one edge of the graph (a boundary edge or the loop) from spectral characteristics, while the coefficients on the other edges are known a priori. We obtain uniqueness theorems and constructive solutions for partial inverse problems.

The inverse problem for differential pencils on a star-shaped graph with mixed spectral data

The partial inverse problem for differential pencils on a star-shaped graph is studied from mixed spectral data. More precisely, we show that if the potentials on all edges on the star-shaped graph but one are known a priori then the unknown potential on the remaining edge can be uniquely determined by partial information on the potential and a part of eigenvalues.

Determination of differential pencils with impulse from interior spectral data

In this paper, we are concerned with the inverse spectral problems for differential pencils defined on [0,pi ] with an interior discontinuity. We prove that two potential functions are determined uniquely by one spectrum and a set of values of eigenfunctions at some interior point bin (0,pi ) in the situation of b=pi /2 and bneq pi /2. For the latter, we need the knowledge of a part of the second spectrum.

The partial inverse nodal problem for differential pencils on a finite interval

In this paper, the partial inverse nodal problem for differential pencils with real-valued coefficients on a finite interval \([0,1]\) was studied. The authors showed that the coefficients \((q_{0}(x),q_{1}(x),h,H_0)\) of the differential pencil \(L_0\) can be uniquely determined by partial nodal data on the right(or, left) arbitrary subinterval \([a,b]\) of \([0,1].\) Finally, an example was given to verify the validity of the reconstruction algorithm for this inverse nodal problem.

Inverse Spectral Problems for Differential Pencils on Arbitrary Compact Graphs

Inverse problems of spectral analysis are studied for second-order differential pencils on arbitrary compact graphs. The uniqueness of recovering operators from their spectra is proved, and a constructive procedure for the solution of this class of inverse problems is provided.

A uniqueness result for differential pencils with discontinuities from interior spectral data

Abstract In this work, the interior spectral data is employed to study the inverse problem for a differential pencil with a discontinuity on the half line. By using a set of values of the eigenfunctions at some internal point and eigenvalues, we obtain the functions q 0 ⁢ ( x ) {q_{0}(x)} and q 1 ⁢ ( x ) {q_{1}(x)} applied in the diffusion operator.

The inverse spectral problem for differential pencils by mixed spectral data

An inverse problem for differential pencils of second order is studied. We show that the potentials on the whole interval can be uniquely determined by partial information on potentials and parts of two spectra.

On the determination of differential pencils with nonlocal conditions

Abstract Inverse problems for differential pencils with nonlocal conditions are considered. Uniqueness theorems of inverse problems from the Weyl-type function and spectra are proved, which are generalizations of the well-known Weyl function and Borg’s inverse problem for the classical Sturm–Liouville operators.

Multiple completeness of the root functions for a certain class of ordinary differential pencils with constant coefficients

A class of polynomial pencils of ordinary differential operators with constant coefficients is considered in the article. The pencils from this class are generated by the n-th order ordinary differential expression and twopoint boundary conditions. Coefficients of the differential expression are supposed to be polynomials of the spectral parameter with constant coefficients. The boundary conditions are supposed to depend on the spectral parameter polynomially. It is assumed that the roots of the characteristic equation of the pencils from this class are simple, non-zero and they are located arbitrarily in the complex plane. The author investigates n-fold completeness of the root functions of the pencils from this class in the space of summable with square functions on the main segment. Sufficient conditions of the n-fold completeness of the root functions are obtained. The main idea of the proof of the theorem is a new asymptotics of the characteristic determinant of the pencil. The presented results supplement previous results of the author.

Inverse problems for differential pencils on A-graphs

Abstract Inverse problems of spectral analysis are studied for differential pencils on A-graphs. The uniqueness of recovering operators from their spectra is proved, and a constructive procedure for the solution of this class of inverse problems is provided.

Inverse spectral problems for differential pencils with the turning point in the finite interval

In this paper, we establish the fundamental system of solutions (FSS) for differential pencils (1.1) by the Birkhoff’s method. We obtain the Weyl function which plays an important role in studying inverse problems. Then we give a formulation of the inverse problem and prove the uniqueness theorem.

О ВОССТАНОВЛЕНИИ ДИФФЕРЕНЦИАЛЬНЫХ ПУЧКОВ НА ГРАФЕ-КУСТЕ

О ВОССТАНОВЛЕНИИ ДИФФЕРЕНЦИАЛЬНЫХ ПУЧКОВ НА ГРАФЕ-КУСТЕ

Inverse spectral problems for differential pencils with boundary conditions dependent on the spectral parameter

In this paper, we discuss two inverse problems for differential pencils with boundary conditions dependent on the spectral parameter. We will prove the Hochstadt–Lieberman type theorem of – except for arbitrary one eigenvalue and the Borg type theorem of – except for at most arbitrary two eigenvalues, respectively. The new results are generalizations of the related results. Copyright © 2016 John Wiley & Sons, Ltd.

Inverse spectral problems for differential pencils on a graph with a rooted cycle

An inverse spectral problem is studied for differential pencils on graphs with a rooted cycle and with standard matching conditions in internal vertices. A uniqueness theorem is proved, and a constructive procedure for the solution is provided.