Spectral Parameter In Boundary Condition Research Articles

OPERATOR-THEORETICAL FORMULATLONS OF BOUNDARY VALUE PROBLEM FOR STURM–LIOUVILLE OPERATOR EQUATlON

Abstract. Our aim in that paper is to set the operator theoretical formulation of boundary value problem for Sturm-Liovuille equation with unbounded operator coefficient and boundary condition involving Herglotz-Nevanlinna function of spectral parameter in boundary condition. We define minimal operator and give description of maximal operator.

Inverse scattering problem for Sturm-Liouville equation with a rational function of spectral parameter in boundary condition

Inverse scattering problem for Sturm-Liouville equation with a rational function of spectral parameter in boundary condition

On the spectral and scattering properties of eigenparameter dependent discrete impulsive Sturm–Liouville equations

This work develops scattering and spectral analysis of a discrete impulsive Sturm-Liouville equation with spectral parameter in boundary condition. Giving the Jost solution and scattering solutions of this problem, we find scattering function of the problem. Discussing the properties of scattering function, scattering solutions, and asymptotic behavior of the Jost solution, we find the Green function, resolvent operator, continuous and point spectrum of the problem. Finally, we give an example in which the main results are made explicit.

Traces and inverse nodal problems for a class of delay Sturm–Liouville operators

In this study, we investigate the regularized sums of eigenvalues, oscillation of eigenfunctions and solutions of inverse nodal problems of discontinuous Sturm-Liouville operators with a delayed argument and with a finite number of transmission conditions. With this aim, we obtain asymptotic formulas for eigenvalues, eigenfunctions and nodal points of the problem. Moreover, some numerical examples are given to illustrate the results. The problem differs from the other discontinuous Sturm-Liouville problems with retarded argument in that it contains a spectral parameter in boundary conditions. If we take the delayed argument $\Delta\equiv0$, the coefficients $\alpha _{i}^{+}=\beta _{i}^{+}=0$ ($i=1,2$) in boundary conditions and the transmission coefficients $\delta_i=1$ ($i=\overline{1,m-1}$) the results obtained below coincide with corresponding results in the classical Sturm-Liouville operator.

On Basis Property of Root Functions For a Class Second Order Differential Operator

Abstract In this work we investigate the completeness, minimality and basis properties of the eigenfunctions of one class discontinuous Sturm-Liouville equation with a spectral parameter in boundary conditions.

Sturm–Liouville Problems Involving Distribution Weights and an Application to Optimal Problems

This paper is concerned with Sturm–Liouville problems (SLPs) with distribution weights and sets up the min–max principle and Lyapunov-type inequality for such problems. As an application, the paper solves the following optimization problems: If the first eigenvalue of a string vibration problem is known, what is the minimal total mass and by which distribution of weight is it attained; if both the first eigenvalue and the total mass are known, what is the corresponding results on the string mass? The vibration problem leads to a SLP with the spectral parameter in both the equation and the boundary conditions. Our main method is to incorporate this problem into the framework of classical SLPs with weights in an appropriate space by transforming it into the one with distribution weight, which provides a different idea for the investigation of the SLPs with spectral parameter in boundary condition.

Solving inverse nodal problem with spectral parameter in boundary conditions

ABSTRACTIn this study, the inverse Sturm-Liouville problem with spectral parameter in boundary conditions on a finite interval is considered. In fact, we use the nodes as input data and compute the approximation of solution of the inverse nodal Sturm-Liouville problem by the first kind Chebyshev polynomials and apply two methods Chebyshev wavelets and Chebyshev interpolation for solving inverse nodal Sturm-Liouville problem. Finally, the results are explained by presenting the numerical example.

On a Boundary Value Problem for a Polynomial Pencil of the Sturm-Liouville Equation with Spectral Parameter in Boundary Conditions

The boundary value problem with a spectral parameter in the boundary conditions for a polynomial pencil of the Sturm-Liouville operator is investigated. Using the properties of the transformation operators for such operators, the asymptotic formulas for eigenvalues of the boundary value problem are obtained.

Inverse spectral problem for Sturm-Liouville operator with discontinuous coefficient and cubic polynomials of spectral parameter in boundary condition

In this paper, the inverse scattering problem for the Sturm-Liouville operator with discontinuous coefficient and cubic polynomials of the spectral parameter in the boundary condition is considered. The scattering data of the problem is defined, and its properties are investigated. The modified Marchenko main equation is obtained and it is shown that the potential is uniquely recovered by the scattering data.

Inverse problem for a class of Sturm-Liouville operator with spectral parameter in boundary condition

This work aims to examine a Sturm-Liouville operator with a piece-wise continuous coefficient and a spectral parameter in boundary condition. The orthogonality of the eigenfunctions, realness and simplicity of the eigenvalues are investigated. The asymptotic formula of the eigenvalues is found, and the resolvent operator is constructed. It is shown that the eigenfunctions form a complete system and the expansion formula with respect to eigenfunctions is obtained. Also, the evolution of the Weyl solution and Weyl function is discussed. Uniqueness theorems for the solution of the inverse problem with Weyl function and spectral data are proved.

Principal Functions of Nonselfadjoint Discrete Dirac Equations with Spectral Parameter in Boundary Conditions

Let L denote the operator generated in ℓ2(ℕ, ℂ2) by , , n∈ℕ, and the boundary condition , where (an) , (bn), (pn), and (qn) , n∈ℕ are complex sequences, γi, βi∈ℂ , i = 0,1, and λ is an eigenparameter. In this paper we investigated the principal functions corresponding to the eigenvalues and the spectral singularities of L.

Inverse Nodal Problem for Dirac System with Spectral Parameter in Boundary Conditions

An inverse nodal problem lies in constructing operators from the given zeros of their eigenfunctions. In this work, we deal with an inverse nodal problem of reconstructing the Dirac system with the spectral parameter in the boundary conditions. We prove that a set of nodal points of one of the components of the eigenfunctions uniquely determines all the parameters of the boundary conditions and the coefficients of the Dirac equations. We also provide a constructive procedure for solving this inverse nodal problem.

On half inverse problem for differential pencils with the spectral parameter in boundary conditions

On half inverse problem for differential pencils with the spectral parameter in boundary conditions

On half inverse problem for differential pencils with the spectral parameter in boundary conditions

A second-order differential pencil on a finite interval with spectral parameter dependent boundary conditions is considered. The inverse problem is studied of recovering the coefficients of the boundary value problem from its spectrum, provided that on one half of the interval they are known a priori. The uniqueness theorem for this inverse problem is proved and a constructive procedure for finding its solution is obtained.

Principal Functions of Non-Selfadjoint Difference Operator with Spectral Parameter in Boundary Conditions

We investigate the principal functions corresponding to the eigenvalues and the spectral singularities of the boundary value problem (BVP) , and , where and are complex sequences, is an eigenparameter, and , for , 1.

A Boundary Value Problem for the Dirac System with a Spectral Parameter in the Boundary Conditions

Problems with a spectral parameter in equations and boundary conditions form an important part of spectral theory of linear differential operators. A bibliography of papers in which such problems were considered in connection with specific physical processes can be found in [1, 2]. Boundary value problems for ordinary differential operators with a spectral parameter in boundary conditions were considered in various settings in numerous papers [3–14]. The completeness and the basis property of eigenfunctions of boundary value problems with a spectral parameter in equations and boundary conditions were studied in more detail in [7, 8]. Consider the following boundary value problem for the Dirac system with the same spectral parameter in the equations and the boundary conditions:

Spectral boundary value problems for the Helmholtz equation with spectral parameter in boundary conditions on a non-smooth surface

The spectral properties of four problems for the Helmholtz equation with spectral parameter in boundary or transmission conditions on a closed Lipschitz surface S are studied. These problems are related to the classical integral operators of potential type on S for the Helmholtz equation. They have been studied before in the case when S is infinitely smooth. It is shown that the most important properties of eigenvalues and root functions hold also for Lipschitz surfaces S. The machinery of potential theory in Lipschitz domains and of spectral theory is used in the proofs.

On operators generated by elliptic boundary problems with a spectral parameter in boundary conditions

General elliptic boundary value problems with the spectral parameter appearing linearly both in the elliptic equation and in boundary conditions are considered. It is proved that the corresponding matrix operator from the Boutet de Monvel algebra is similar to an almost diagonal operator. This result is applied to prove the completeness and the summability (in the sense of Abel) of the root vectors of this operator.

Equivalence of a part of derived chains of boundary-value problems for second-order ordinary differential equations

A method is proposed for obtaining criteria for the equivalence of a part of derived chains of boundary-value problems for second-order differential equations with a spectral parameter in boundary conditions.