# Discrete Sturm-Liouville problems with parameter in the boundary conditions

B. J. Harmsen, A. Li

Research output: Contribution to journalArticle

7 Citations (Scopus)

### Abstract

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.

Original language English 969-981 13 Journal of Difference Equations and Applications 8 11 https://doi.org/10.1080/1023619021000048869 Published - 1 Nov 2002

### Fingerprint

Eigenvalues and Eigenvectors
Sturm-Liouville Problem
Eigenvalues and eigenfunctions
Boundary value problems
Boundary Value Problem
Boundary conditions
Restriction
Eigenvalue
Green's Formula
Sturm-Liouville Equation
Linear algebra
Difference equations
Difference equation
System of equations
Linear systems
Linearly
Linear Systems
Distinct
Coefficient

### Keywords

• Boundary value problem
• Difference equation
• Discrete Sturm-Liouville problem
• Eigenvalue
• Parameter in the boundary conditions

### Cite this

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abstract = "This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.",
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In: Journal of Difference Equations and Applications, Vol. 8, No. 11, 01.11.2002, p. 969-981.

Research output: Contribution to journalArticle

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AB - This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.

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KW - Difference equation

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