scholarly journals A Sturm-Liouville Problem with a Discontinuous Coefficient and Containing an Eigenparameter in the Boundary Condition

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Erdoğan Şen
Keyword(s):  
Boundary Condition ◽  
Green Function ◽  
Boundary Conditions ◽  
Liouville Operator ◽  
Liouville Problem ◽  
Fundamental Solutions ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville ◽  
Interior Points

We study a Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points. We give an operator-theoretic formulation, construct fundamental solutions, investigate some properties of the eigenvalues and corresponding eigenfunctions of the discontinuous Sturm-Liouville problem and then obtain asymptotic formulas for the eigenvalues and eigenfunctions and find Green function of the discontinuous Sturm-Liouville problem.

Sturm-Liouville Problems with Discontinuities at Two Interior Points

2018 ◽  
Vol 85 (1-2) ◽  
pp. 70
Author(s):  
Hongmei Han
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Liouville Operator ◽  
Resolvent Operator ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville ◽  
Basic Solutions ◽  
The Resolvent Operator ◽  
Interior Points

<p>In this paper, we study the Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points. We establish a new operator <em>A</em> associated with the problem, prove the operator <em>A</em> is self-adjoint in an appropriate space <em>H</em>, construct the basic solutions and investigate some properties of the eigenvalues and corresponding eigenfunctions, then obtain asymptotic formulas for the eigenvalues and eigenfunctions, its Green function and the resolvent operator are also involved.</p>

scholarly journals Asymptotic Behaviour of Eigenvalues and Eigenfunctions of a Sturm-Liouville Problem with Retarded Argument

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Erdoğan Şen ◽  
Jong Jin Seo ◽  
Serkan Araci
Keyword(s):  
Boundary Value Problem ◽  
Liouville Operator ◽  
Liouville Problem ◽  
Asymptotic Formulas ◽  
Retarded Argument ◽  
Eigenvalues And Eigenfunctions ◽  
Transmission Coefficients ◽  
Discontinuous Boundary ◽  
Sturm Liouville ◽  
Special Case

In the present paper, a discontinuous boundary-value problem with retarded argument at the two points of discontinuities is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. This is the first work containing two discontinuities points in the theory of differential equations with retarded argument. In that special case the transmission coefficients and retarded argument in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.

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

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jinming Cai ◽  
Zhaowen Zheng
Keyword(s):  
Boundary Conditions ◽  
Liouville Problem ◽  
Fundamental Solutions ◽  
Asymptotic Formulas ◽  
Limit Circle ◽  
Operator Formulation ◽  
Sturm Liouville ◽  
Completeness Of Eigenfunctions

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.

scholarly journals ON THE EXPANSION FORMULA FOR A SINGULAR STURM-LIOUVILLE OPERATOR

2021 ◽  
Vol 21 (1) ◽  
pp. 67-76
Author(s):  
ULVIYE DEMIRBILEK ◽  
KHANLAR R. MAMEDOV
Keyword(s):  
Boundary Condition ◽  
Green Function ◽  
Spectral Parameter ◽  
Liouville Operator ◽  
Resolvent Operator ◽  
Liouville Problem ◽  
Scattering Data ◽  
Expansion Formula ◽  
Sturm Liouville

In this study, on the semi-axis, Sturm - Liouville problem under boundary condition depending on spectral parameter is considered. In what follows scattering data is defined and its properties are given for the problem. The kernel of resolvent operator which is Green function is constructed. Using Titchmarsh method, expansion is obtained according to eigenfunctions and expansion formula is expressed with the scattering data.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

scholarly journals Fundamental Spectral Theory of Fractional Singular Sturm-Liouville Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Erdal Bas
Keyword(s):  
Spectral Theory ◽  
Liouville Operator ◽  
Spectral Properties ◽  
Liouville Problem ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville

We give the theory of spectral properties for eigenvalues and eigenfunctions of Bessel type of fractional singular Sturm-Liouville problem. We show that the eigenvalues and eigenfunctions of the problem are real and orthogonal, respectively. Furthermore, we prove new approximations about the topic.

Partial inverse problems for the Sturm–Liouville operator on a star-shaped graph with mixed boundary conditions

2018 ◽  
Vol 26 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Natalia Pavlovna Bondarenko
Keyword(s):  
Inverse Problems ◽  
Boundary Conditions ◽  
Liouville Operator ◽  
Mixed Boundary ◽  
Constructive Method ◽  
Asymptotic Formulas ◽  
Riesz Basis Property ◽  
Partial Inverse ◽  
Sturm Liouville ◽  
Shaped Graph

AbstractThe Sturm–Liouville operator on a star-shaped graph with different types of boundary conditions (Robin and Dirichlet) in different vertices is studied. Asymptotic formulas for the eigenvalues are derived and partial inverse problems are solved: we show that the potential on one edge can be uniquely determined by different parts of the spectrum if the potentials on the other edges are known. We provide a constructive method for the solution of the inverse problems, based on the Riesz basis property of some systems of vector functions.

Expansions in terms of sets of functions with complex eigenvalues

1948 ◽  
Vol 44 (2) ◽  
pp. 242-250 ◽  
Author(s):  
R. Peierls
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Real Function ◽  
Liouville Problem ◽  
Nuclear Collisions ◽  
Complex Eigenvalues ◽  
Sturm Liouville ◽  
Image Position ◽  
Unusual Type

In the following I discuss the properties, in particular the completeness of the set of eigenfunctions, of an eigenvalue problem which differs from the well-known Sturm-Liouville problem by the boundary condition being of a rather unusual type.The problem arises in the theory of nuclear collisions, and for our present purpose we take it in the simplified formwhere 0 ≤ x ≤ 1. V(x) is a given real function, which we assume to be integrable and to remain between the bounds ± M, and W is an eigenvalue. The eigenfunction ψ(x) is subject to the boundary conditionsand

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