Discontinuous Sturm-Liouville Problems Research Articles

THE REGULARIZED TRACE FORMULA FOR A DISCONTINUOUS STURM-LIOUVILLE PROBLEM WITH A SPECTRAL PARAMETER DEPENDENT JUMP CONDITION

The first regularized trace formula for a discontinuous Sturm-Liouville problem with a spectral parameter dependent jump condition is obtained.

Eigenvalue properties of Sturm-Liouville problems with transmission conditions dependent on the eigenparameter

<abstract><p>This paper studies a discontinuous Sturm-Liouville problem in which the spectral parameter appears not only in the differential equation but also in the transmission conditions. By constructing an appropriate Hilbert space and inner product, the eigenvalue and eigenfunction problems of the Sturm-Liouville problem are transformed into an eigenvalue problem of a certain self-adjoint operator. Next, the eigenfunctions of the problem and some properties of the eigenvalues are given via construction of the basic solution. The Green's function for the Sturm-Liouville problem is also given. Finally, the continuity of the eigenvalues and eigenfunctions of the problem is discussed. Especially, the differential expressions of the eigenvalues for some parameters have been obtained, including the parameters in the eigenparameter-dependent transmission conditions.</p></abstract>

Spectral Properties of Discontinuous Boundary Value Problem with Transmission Conditions and Manypoint Singularities

In this study, we investigate a discontinuous Sturm-Liouville boundary value problem on three intervals
 with manypoint-transmission conditions in direct sum of Sobolev space. We establish such spectral properties as Fredholmness and coreciveness with respect to the eigenvalue parameter

EIGENVALUES OF STURM-LIOUVILLE PROBLEMS WITH EIGENPARAMETER DEPENDENT BOUNDARY AND INTERFACE CONDITIONS

In this paper, a regular discontinuous Sturm-Liouville problem which contains eigenparameter in both boundary and interface conditions is investigated. Firstly, a new operator associated with the problem is constructed, and the self-adjointness of the operator in an appropriate Hilbert space is proved. Some properties of eigenvalues are discussed. Finally, the continuity of eigenvalues and eigenfunctions is investigated, and the differential expressions in the sense of ordinary or Fréchet of the eigenvalues concerning the data are given.

A generalized discontinuous Sturm–Liouville problem with boundary conditions rationally dependent on the eigenparameter

The present paper deals with a class of generalized discontinuous Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter. Using operator theoretic formulation under the new inner product based on boundary conditions rationally dependent on the eigenparameter and general transmission conditions, the self-adjointness of this operator is proved. We also 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 general transmission conditions, as well as the endpoints. Moreover, we find the differential expressions for each parameters, respectively.

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.

Uniqueness Theorems for Inverse Problems of Discontinuous Sturm–Liouville Operator

In this paper, inverse spectral problems of discontinuous Sturm–Liouville problems contained in the discontinuous coefficient and discontinuity conditions at an interior point of the finite interval are, respectively, studied according to

Bounds for distance between eigenvalues of boundary value problems with retarded argument

Abstract In this study we are concerned with spectrum of boundary value problems with retarded argument with discontinuous weight function, two supplementary transmission conditions at the point of discontinuity, spectral and physical parameters in the boundary condition and we obtain bounds for the distance between eigenvalues. We extend and generalize some approaches and results of the classical regular and discontinuous Sturm-Liouville problems. In the special case that ω (x) ≡ 1, the transmission coefficients γ 1 = δ 1, γ 2 = δ 2 and retarded argument Δ ≡ 0 in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.

Discontinuous fractional Sturm–Liouville problems with transmission conditions

The main purpose of this study is the investigation of discontinuous Sturm–Liouville problem with fractional derivatives. We give an operator theoretic framework of the problem under consideration. Namely, we define an operator A in the Hilbert space L2[−1,1], the eigenvalues and corresponding eigenfunctions of which coincide with the eigenvalues and corresponding eigenfunctions of the boundary value problem, respectively. Then, we establish the characteristic function and prove that the eigenvalues of the considered problem coincide with the roots of this characteristic function.

Computation of Trace and Nodal Points of Eigenfunctions for a Sturm-Liouville Problem with Retarded Argument

In this study, a formula for regularized sums of eigenvalues and nodal points of eigenfunctions for a discontinuous Sturm-Liouville problem with a constant retarded argument. Contrary to standart problems the spectral parameter appears not only in the differential equation, but also in one of the boundary conditions. Thus, we see whether the nodal points of eigenfunctions and the trace change or not.

Stability of the Reconstruction Discontinuous Sturm-Liouville Problem

In this work, we study stability of the inverse spectral problem for the Sturm-Liouville operator -D²+q with discontinuity boundary conditions inside a finite closed interval. We use the method which is given by Ryabushko for regular Sturm-Liouville operator in ryb to obtain stability results. These results give a bound for the difference between the spectral functions of associated problems. In addition, we give asymptotic representation of the eigenvalues and a formula for the representation of the norming constants by two spectra.

Inverse spectral problems for discontinuous Sturm–Liouville problems of Atkinson type

We investigate inverse spectral problems for discontinuous Sturm–Liouville problems of Atkinson type whose spectrum consists of a finite set of eigenvalues. For given two finite sets of interlacing real numbers, there exists a class of Sturm–Liouville equations such that the two sets of numbers are exactly the eigenvalues of their associated Sturm–Liouville problems with two different separated boundary conditions. The main approach is to give an equivalent relation between Sturm–Liouville problems of Atkinson type and matrix eigenvalue problems, and the theory of inverse matrix eigenvalue problems.

MINIMIZATION PRINCIPLE AND GENERALIZED FOURIER SERIES FOR DISCONTINUOUS STURM-LIOUVILLE SYSTEMS IN DIRECT SUM SPACES

By modifing the Green's function method we study certain spectral aspects of discontinuous Sturm-Liouville problems with interior singularities. Firstly, we define four eigen-solutions and construct the Green's function in terms of them. Based on the Green's function we establish the uniform convergeness of generalized Fourier series as eigenfunction expansion in the direct sum of Lebesgue spaces L2 where the usual inner product replaced by new inner product. Finally, we extend and generalize such important spectral properties as Parseval equation, Rayleigh quotient and Rayleigh-Ritz formula (minimization principle) for the considered problem.

On the fundamental solutions of a discontinuous fractional boundary value problem

The main purpose of this study is to investigate a fractional discontinuous Sturm-Liouville problem with transmission conditions. We shall consider a fractional boundary value problem involving an operator with two parts. It is shown that the eigenvalues and corresponding eigenfunctions of the main problem coincide with the eigenvalues and corresponding eigenfunctions of the constructed operator in Hilbert spaces.

The Basis Property of the System of Weak Eigenfunctions of a Discontinuous Sturm–Liouville Problem

The purpose of this study is to investigate some important spectral properties of one discontinuous Sturm–Liouville problem. By applying our own approaches the considered problem is transformed into an eigenvalue problem for suitable integral equation in terms of which it is defined as a concept of weak eigenfunctions. Then we introduce for consideration some compact operators in such a way that this integral equation can be reduced to the appropriate operator-pencil equation and prove that this operator-pencil is self-adjoint and positive definite for sufficiently large negative values of the eigenparameter. Finally, it is established that the spectrum is discrete and the system of corresponding weak eigenfunctions forms an orthonormal basis of the appropriate Hilbert space.

Spectral Problem for the Sturm–Liouville Operator with Retarded Argument Containing a Spectral Parameter in the Boundary Condition

We consider a discontinuous Sturm–Liouville problem with retarded argument containing a spectral parameter in the boundary condition. First, we investigate the simplicity of eigenvalues and then prove the existence theorem. As a result, we obtain the asymptotic formulas for eigenvalues and eigenfunctions.

A trace formula for the Sturm-Liouville type equation with retarded argument

In this paper, we deal with a discontinuous Sturm-Liouville problem with retarded argument and eigenparameter-dependent boundary conditions. We obtain the asymptotic formulas for the eigenvalues and the regularized trace formula for the problem

A Singular Sturm-Liouville Problem with Limit Circle Endpoints and Eigenparameter Dependent Boundary Conditions

In this paper, we investigate a class of discontinuous singular Sturm-Liouville problems with limit circle endpoints and eigenparameter dependent boundary conditions. Operator formulation is constructed and asymptotic formulas for eigenvalues and fundamental solutions are given. Moreover, the completeness of eigenfunctions is discussed.

Completeness of eigenfunctions of discontinuous Sturm–Liouville problems

In this work we investigate the resolvent operator and completeness of eigenfunctions of a Sturm–Liouville problem with discontinuities at two points. The problem contains an eigenparameter in one of the boundary conditions. For operator-theoretic formulation of the considered problem we define an equivalent inner product in the Hilbert space \( L_{2} [ - 1,1] \oplus C \) and suitable self-adjoint linear operator in it.

Sturm-Liouville problem with moving discontinuity points

In this paper, we present a new discontinuous Sturm Liouville problem with symmetrically located discontinuities which are defined depending on a neighborhood of a midpoint of the interval. Also the problem contains an eigenparameter in one of the boundary conditions and has coupled transmission conditions at the discontinuity points. We investigate the properties of the eigenvalues, obtain asymptotic formulas for the eigenvalues and the corresponding eigenfunctions and construct Green's function of this problem.