Eigenparameter Dependent Boundary Conditions Research Articles

Inverse problem for Sturm–Liouville operator with complex-valued weight and eigenparameter dependent boundary conditions

Abstract This paper is concerned with discontinuous inverse problem generated by complex-valued weight Sturm–Liouville differential operator with λ-dependent boundary conditions. We establish some properties of spectral characteristic and prove that the potential on the whole interval can be uniquely determined by the Weyl-type function or two spectra.

Solutions to a discrete resonance problem with eigenparameter-dependent boundary conditions

<abstract><p>In this paper, we considered the existence of solutions to a discrete second-order resonance problem with eigenparameter-dependent boundary conditions. We first transformed the resonance problem into its corresponding equivalent system using the Lyapunov-Schmidt method. In addition, using Schauder's fixed-point theorem and the connectivity theories of the solution set of compact vector fields, we obtained the existence and multiplicity of solutions to the second-order discrete resonance problem with eigenparameter-dependent boundary conditions.</p></abstract>

Sturm-Liouville Problems with Polynomially Eigenparameter Dependent Boundary Conditions

Sturm-Liouville equation on a finite interval together with boundary conditions arises from the infinitesimal, vertical vibrations of a string with the ends subject to various constraints. The coefficient (also called potential) function in the differential equation is in a close relationship with the density of the string. In this sense, the computation of solutions plays a rather important role in both mathematical and physical fields. In this study, asymptotic behaviors of the solutions for Sturm-Liouville problems associated with polynomially eigenparameter dependent boundary conditions are obtained when the potential function is real valued 𝑳𝟏- function on the interval (𝟎, 𝟏). Besides, the asymptotic formulae are given for the derivatives of the solutions.

An inverse scattering problem for eigenparameter-dependent discrete Dirac system with Levinson formula

In this paper, we have considered an inverse scattering problem for discrete Dirac operator L with eigenparameter dependent boundary condition. At the outset, the Jost solution and the Jost function of L are given. Then the zeros of the Jost function and some properties of the scattering function are examined. After finding the scattering dataset and the main equation for the operator L, the uniqueness of the kernel, continuity of the scattering function and the appropriate Levinson type formula are obtained.

Asymptotic Estimations of Eigenvalues and Eigenfunctions for Nonlocal Boundary Value Problems with Eigenparameter-Dependent Boundary Conditions

The nonlocal boundary value problem with eigenparameter dependent boundary conditions is studied in this work. Firstly, we give the asymptotic expressions of the general solution for the equation corresponding to the initial conditions and prove the simplicity of eigenvalues. Moreover, the general properties of the eigenvalues and eigenfunctions for such a problem are proved. Finally, the asymptotic formulas of eigenvalues and eigenfunctions are obtained under certain mild conditions. Our method is to incorporate the perturbation theory and asymptotic analysis in the framework of classical Sturm-Liouville problems, which provides a different idea for the investigating of the Sturm-Liouville problems with eigenparameter in boundary conditions.

Inverse square singularities and eigenparameter-dependent boundary conditions are two sides of the same coin

Abstract We show that inverse square singularities can be treated as boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter with ‘a negative number of poles’. More precisely, we treat in a unified manner one-dimensional Schrödinger operators with either an inverse square singularity or a boundary condition containing a rational Herglotz–Nevanlinna function of the eigenvalue parameter at each endpoint and define Darboux-type transformations between such operators. These transformations allow one, in particular, to transfer almost any spectral result from boundary value problems with eigenparameter-dependent boundary conditions to those with inverse square singularities and vice versa.

Inverse problems for discontinuous Dirac operator with eigenparameter dependent boundary and transmission conditions

AbstractIn this study, we consider the discontinuous Dirac equations system with eigenparameter dependent boundary and finite number of transmission conditions. First, the space that corresponds to problem is introduced, the norm on this space is defined and the operator model that corresponds to the given problem is constructed on this space. Then the integral equations and asymptotics of eigenfunctions of the problem are obtained. The characteristic function is defined and the asymptotic formula of the characteristic function is given by using obtained asymptotics of eigenfunctions. After the Weyl solution and the Weyl function of the problem are formed. Finally, some uniqueness theorems are proved by using Weyl function and some spectral data.

Spectral analysis of discontinuous Sturm-Liouville operators with Herglotzs transmission

<abstract><p>In this paper, we study the spectral properties of the Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions. In details, we introduce a Hilbert space formula, so that the problem we consider can be interpreted as an eigenvalue problem of an self-adjoint operator. Moreover, the Green's function and the resolvent of the related linear operator are obtained.</p></abstract>

A Class of Singular Sturm–Liouville Problems with Discontinuity and an Eigenparameter-Dependent Boundary Condition

In this paper, we study a singular Sturm–Liouville problem with an eigenparameter-dependent boundary condition and transmission conditions at two interior points. Using an operator-theoretical formulation, we transfer the problem to an operator in an appropriate Hilbert space. It is proved that the operator is self-adjoint. We also give the asymptotic formulas of the eigenvalues of the problem. Moreover, Green’s function is also discussed.

Dependence of Eigenvalues of Discontinuous Fourth-order Differential Operators with Eigenparameter Dependent Boundary Conditions

In this paper, we investigate a fourth-order differential operator with eigenparameter dependent boundary conditions and transmission conditions. To study the eigenvalues of the problem, we establish a new operator associated with the considered problem. Furthermore, we prove that the eigenvalues are differentiable depending on the parameters of the problem. Finally, the differential expressions of the eigenvalues with respect to all parameters are given.

An examination of boundary value transmission problem with quadratic spectral parameter

The main purpose of this study is to examine the properties of scattering solutions and scattering function of a difference equation subject to eigenparameter-dependent boundary and transmission conditions. In general, we are interested in a boundary value problem (BVP) generated by this difference equation, eigenparameter-dependent boundary and transmission conditions. Introducing the Jost solution and scattering solutions of BVP, we obtain the scattering function of this problem. Also, we find the resolvent operator, eigenvalues and continuous spectrum of the operator generated by this BVP. At the end, we present an example to demonstrate the effectiveness of our main results.

Dependence of Eigenvalues of Sturm–Liouville Problems with Eigenparameter-Dependent Boundary Conditions and Interface Conditions

In this paper, regular Sturm–Liouville problems with eigenparameter-dependent boundary conditions and interface conditions are investigated. We obtain that the eigenvalues of the problems depend not only continuously but also smoothly on the parameters of the problem: the coefficients, the boundary conditions, the interface conditions, as well as the endpoints. Moreover, we find the differential expressions for each parameters, respectively.

Eigenvalues of a Class of Eigenparameter Dependent Third-Order Differential Operators

The present paper deals with a class of third-order differential operators with eigenparameter dependent boundary conditions. The continuity of eigenvalues concerning a given parameter is proved. Moreover, the derivative formulas of eigenvalues concerning the parameters, in particular, the eigenparameter dependent boundary condition matrix, are given.

Eigenvalues of Sturm-Liouville problems with distributional potentials and eigenparameter-dependent boundary conditions

In this paper, regular Sturm-Liouville problems with distributional potentials and eigenparameter-dependent boundary conditions are investigated. We obtain that the eigenvalues of the problems depend not only continuously but also smoothly on the parameters of the problem: the coefficients, the boundary conditions, and the endpoints, in particular, the eigenparameter-dependent boundary condition matrix. Moreover, we find the differential expressions for each parameters.

Eigenvalues of fourth-order differential operators with eigenparameter dependent boundary conditions

<abstract><p>This paper is concerned with a fourth-order differential operator with eigenparameter dependent boundary conditions. We prove that each of the eigenvalues of the problem can be embedded in a continuous eigenvalue branch. Furthermore, the differential expressions of the eigenvalues with respect to each of parameters are given.</p></abstract>

Properties for fourth order discontinuous differential operators with eigenparameter dependent boundary conditions

<abstract><p>In this paper, a class of fourth order differential operators with eigenparameter-dependent boundary conditions and transmission conditions is considered. A new operator associated with the problem is established, and the self-adjointness of this operator in an appropriate Hilbert space $ H $ is proved. The fundamental solutions are constructed. Sufficient and necessary conditions of the eigenvalues are investigated. Then asymptotic formulas for the fundamental solutions and the characteristic functions are given. Finally, the completeness of eigenfunctions in $ H $ is given and the Green function is also involved.</p></abstract>

Spectral problems for Sturm-Liouville operator with eigenparameter boundary conditions on time scales

In this paper, we consider the inverse problem for Sturm-Liouville operators with eigenparameter dependent boundary conditions on time scales. We give new uniqueness theorems and investigate its some special cases.

The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions

Abstract For any positive integer n and a set of positive integers m i {m}_{i} , i = 1 , 2 , … , n + 1 i=1,2,\ldots ,n+1 , we construct a class of quadratic eigenparameter-dependent boundary Sturm-Liouville problems with n transmission conditions, which have at most ∑ i = 1 n + 1 m i + n + 5 {\sum }_{i=1}^{n+1}{m}_{i}+n+5 eigenvalues. The key to this analysis is still the division of intervals and an iterative construction of the characteristic function. Further, some examples are given for a simple explanation.

Global behavior and nontrivial solutions for discrete Sturm–Liouville problems with eigenparameter-dependent boundary condition

Global behavior and nontrivial solutions for discrete Sturm–Liouville problems with eigenparameter-dependent boundary condition

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.