Eigenparameter Dependent Boundary Conditions Research Articles

Eigenvalues of discontinuous third-order boundary value problems with eigenparameter-dependent boundary conditions

Regular third-order boundary value problems with eigenparameter-dependent boundary conditions and transmission conditions are investigated. It is obtained that the eigenvalues of the considered problems depend not only continuously but also smoothly on the parameters of the problems, i.e., the coefficients, the boundary conditions and transmission conditions. Moreover, the corresponding differential expressions for each parameters are given.

DISCONTINUOUS FRACTIONAL STURM-LIOUVILLE PROBLEMS WITH EIGEN-DEPENDENT BOUNDARY CONDITIONS

<abstract> In this paper, a fractional discontinuous Sturm-Liouville type boun-dary-value problem with eigenparameter-dependent boundary conditions and with two fractional transmission conditions is investigated. Using operator theory, a new inner product is defined by combining the parameters in the boundary and transmission conditions, then the boundary value transmission problem is transferred to an operator in a new Hilbert space such that the eigenvalues and eigenfunctions of the main problem coincide with those of this operator. Moreover, the fundamental solutions are constructed, and then the characteristic function whose zeros are the eigenvalues of the problem is established. </abstract>

On Inverse Nodal Problem and Multiplicities of Eigenvalues of a Vectorial Sturm-Liouville Problem

An m-dimensional vectorial inverse nodal Sturm-Liouville problem with eigenparameter-dependent boundary conditions is studied. We show that if there exists an infinite sequence ynj,rx,λnj,r2j=1∞ of eigenfunctions which are all vectorial functions of type (CZ), then the potential matrix Qx and A are simultaneously diagonalizable by the same unitary matrix U. Subsequently, some multiplicity results of eigenvalues are obtained.

On the Hochstadt–Lieberman type problem with eigenparameter dependent boundary condition

Abstract The half-inverse problem is studied for the Sturm–Liouville operator with an eigenparameter dependent boundary condition on a finite interval. We develop a reconstruction procedure and prove the existence theorem for solution of the inverse problem. Our method is based on interpolation of entire functions.

Dependence of eigenvalues of Sturm–Liouville problems with eigenparameter dependent boundary conditions

This paper is concerned with regular Sturm–Liouville problems with eigenparameter dependent boundary conditions. We obtain that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem, in particular, eigenparameter-dependent boundary condition matrix. Moreover the differential expression of the eigenvalues with respect to the data is given. The dependence of the nth eigenvalue as a function of the parameters of the problem is also investigated by Pru¨fer transformation and the inequalities of eigenvalues.

A Sturm-Liouville Type Problem with Retarded Argument Which Contains a Spectral Parameter in the Boundary Condition

The aim of this study is to investigate a discontinuous problem of the Sturm-Liouville type with a retarded argument containing a spectral parameter in the boundary condition and two additional conditions of transmission at the two discontinuity points. Also, eigenparameter dependent boundary conditions and obtains asymptotic formulas for the eigenvalues and eigenfunctions. And the spectral parameter is real. And the real valued function is continuous on that interval and it has a finite limit. The goal of this article is to obtain asymptotic formulas for eigenvalues of eigenfunctions for problem of the form : with boundary conditions and transmission conditions To this aim, first, the principal term of asymptotic distribution of eigenvalues and eigenfunctions of given problem was obtained up to but, afterwards under some additional conditions, we improve these formulas up to Thus, when the number of points of discontinuity is more than one, we see how the asymptotic behaviour of eigenvalues and eigenfunctions of a boundary value problem with retarded argument which contains a spectral parameter in the boundary conditions change.

Completeness Theorem for Eigenparameter Dependent Dissipative Dirac Operator with General Transfer Conditions

This paper deals with a singular (Weyl’s limit circle case) non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition and finite general transfer conditions. Using the equivalence between Lax-Phillips scattering matrix and Sz.-Nagy-Foiaş characteristic function, the completeness of the eigenfunctions and associated functions of this dissipative operator is discussed.

Inverse spectral problem of a class of fourth-order eigenparameter-dependent boundary value problems

This paper deals with a class of inverse spectral problems of fourth-order boundary value problems with eigenparameter-dependent boundary conditions. Under the equivalent conditions of the problem and a certain type of matrix eigenvalue problem some coefficient functions are reconstructed from the given three sets of interlacing real numbers and several additional conditions. The key technique is the method of inverse matrix eigenvalue problems of a two-banded matrix.

Regular approximation of singular Sturm\u2013Liouville problems with eigenparameter dependent boundary conditions

In this paper we consider singular Sturm–Liouville problems with eigenparameter dependent boundary conditions and two singular endpoints. The spectrum of such problems can be approximated by those of the inherited restriction operators constructed. Via the abstract operator theory, the strongly resolvent convergence and norm resolvent convergence of a sequence of operators are obtained and it follows that the spectral inclusion of spectrum holds. Moreover, spectral exactness of spectrum holds for two special cases.

О БАЗИСНОМ СВОЙСТВЕ КОРНЕВЫХ ФУНКЦИЙ ОДНОГО КЛАССА ДИФФЕРЕНЦИАЛЬНЫХ ОПЕРАТОРОВ ВТОРОГО ПОРЯДКА

It is well known that the Sturmian the ory is an important tool in solving numerous problems of mathematical physics. Usually, eigenvalue parameter appears linearly only in the differential equation of the classic Sturm–Liouville problems. However, in mathematical physics there are also problems, which contain eigenvalue parameter not only in differential equation, but also in the boundary conditions. In this paper, we consider a Sturm–Liouville equation with the eigenparameter dependent boundary condition and with transmission conditions at two points of discontinuity. The aim of this paper is to investigate the completeness, minimality and basis properties of rootfunctions for the considered boundary value problem.

Spectra of a discrete Sturm-Liouville problem with eigenparameter-dependent boundary conditions in Pontryagin space

In this paper, we consider the spectra of a discrete Sturm-Liouville problem with two eigenparameter-dependent boundary conditions. Different from the previous results, the operator corresponding to the problem here is not self-adjoint in the corresponding Hilbert space, but J-self-adjoint in a new Pontryagin space under a new J-metric. No matter what, the existence, simplicity and interlacing properties of real eigenvalues, nonreal eigenvalues and the oscillation properties of eigenfunctions are obtained in this paper. Meanwhile, we also obtain the existence of a pair of nonreal eigenvalues and an unconditional basis of the Pontryagin space under the new J-metric.

On a class of inverse Sturm–Liouville problems with eigenparameter-dependent boundary conditions

A class of inverse Sturm–Liouville problems of Atkinson type with eigenparameter-dependent boundary conditions is investigated. By using the matrix representations of the Sturm–Liouville problems with eigenparameter-dependent boundary conditions and a special class of inverse matrix eigenvalue problems, the corresponding Sturm–Liouville problems are constructed by using two sets of interlacing real numbers satisfying certain conditions. An algorithm and some examples are also given to illustrate our results.

Spectral Properties of Discrete Klein–Gordon s-Wave Equation with Quadratic Eigenparameter-Dependent Boundary Condition

In this study, we consider the spectral analysis of the boundary value problem (BVP) consisting of the discrete Klein–Gordon equation and the quadratic eigenparameter-dependent boundary condition. Presenting the Jost solution and Green’s function, we investigate the finiteness and other spectral properties of the eigenvalues and spectral singularities of this BVP under certain conditions.

Reconstruction of potential and boundary conditions for second order difference equations

Assume the eigenvalues and the weights are given for a difference boundary value problem and that the form of the boundary conditions at the endpoints is known. In particular, it is known whether the endpoints are fixed (i.e. Dirichlet or non-Dirichlet boundary conditions) or whether the endpoints are free to move (i.e. boundary conditions with affine dependence on the eigenparameter). This work illustrates how the potential as well as the exact boundary conditions can be uniquely reconstructed. The procedure is inductive on the number of unit intervals. This paper follows along the lines of S. Currie and A. Love, Inverse problems for difference equations with quadratic eigenparameter dependent boundary conditions, \emph{Quaestiones Mathematicae}, 40 (2017), no. 7, 861−877. Since the inverse problem considered in this paper contains more unknowns than the inverse problem considered in the above reference, an additional spectrum is required more often than was the case in the unique reconstruction of the potential alone.

Inverse spectral problem for Sturm-Liouville operator with coupled eigenparameter-dependent boundary conditions of Atkinson type

ABSTRACTThis paper is devoted to the inverse spectral theory for Sturm-Liouville problems of Atkinson type with coupled eigenparameter-dependent boundary conditions. By using the so called matrix representations of the Sturm-Liouville problems with coupled eigenparameter-dependent boundary conditions and certain type of inverse matrix eigenvalue problems, the corresponding Sturm-Liouville problems are constructed by using two sets of interlacing real numbers and certain conditions.

A note on the spectrum of discrete Klein-Gordon s-wave equation with eigenparameter dependent boundary condition

This paper is concerned with the boundary value problem (BVP) for the discrete Klein-Gordon equation ?(an-1?yn-1)+(vn-?)2 yn = 0; n ? N and the boundary condition (?0+?1?)y1+(?0+?1)y0 = 0 where (an),(vn) are complex sequences, ?i, ?i ? C, i=0,1 and ? is a eigenparameter. The paper presents Jost solution, eigenvalues, spectral singularities and states some theorems concerning quantitative properties of the spectrum of this BVP under the condition ?n?N exp(?n?)(|1-an| + |vn|) &lt; ? for ? &gt; 0 and 1/2 ? ? ? 1.

A discontinuous q-fractional boundary value problem with eigenparameter dependent boundary conditions

A discontinuous q-fractional boundary value problem with eigenparameter dependent boundary conditions

Computing Structured Singular Values for Sturm-Liouville Problems

In this article we present numerical computation of pseudo-spectra and the bounds of Structured Singular Values (SSV) for a family of matrices obtained while considering matrix representation of SturmLiouville (S-L) problems with eigenparameter-dependent boundary conditions. The low rank ODE's based technique is used for the approximation of the bounds of SSV. The lower bounds of SSV discuss the instability analysis of linear system in system theory. The numerical experimentation show the comparison of bounds of SSV computed by low rank ODE'S technique with the well-known MATLAB routine mussv available in MATLAB Control Toolbox.

Finite spectrum of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on time scales

The spectral analysis of a class of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on bounded time scales is investigated. By partitioning the bounded time scale such that the coefficients of Sturm-Liouville equation satisfy certain conditions on the adjacent subintervals, the finite eigenvalue results are obtained. The results show that the number of eigenvalues not only depend on the partition of the bounded time scale, but also depend on the eigenparameter-dependent boundary conditions. Both of the self-adjoint and non-self-adjoint cases are considered in this paper.

On the solution of an inverse Sturm–Liouville problem with a delay andeigenparameter-dependent boundary conditions

In this paper, a boundary value problem consisting of a delay differential equation of the Sturm-Liouville type with eigenparameter-dependent boundary conditions is investigated. The asymptotic behavior of eigenvalues is studied and the parameter of delay is determined by eigenvalues. Then we obtain the connection between the potential function and the canonical form of the characteristic function.