The Completeness of Eigenfunctions

2010 ◽  
pp. 174-209
Keyword(s):  
Completeness Of Eigenfunctions

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.

Multiparameter Sturm theory

1984 ◽  
Vol 99 (1-2) ◽  
pp. 173-184 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne
Keyword(s):  
Elliptic Partial Differential Equation ◽  
Comparison Theorems ◽  
Unified Theory ◽  
Focal Points ◽  
The Matrix ◽  
Sturm Liouville ◽  
Left And Right ◽  
Image Position ◽  
Completeness Of Eigenfunctions ◽  
Sturm Theory

SynopsisLet Sturm–Liouville problemswith continuous coefficients and appropriate boundary conditions, be coupled by the eigenvalue λ = (λ1, … λk). When k = 1, there are various oscillation, perturbation and comparison theorems concerning existence and continuous or monotonic dependence of eigenvalues, eigenfunctions and their zeros (i.e. focal points).We attempt a unified theory for such results, valid for general fc, under conditions known as "left" and “right” definiteness. A representative result may be stated loosely as follows: if LD holds then (elementwise) monotonic dependence of p, q and the matrix [ars] forces monotonic dependence of λ. LD is a generalisation of the “polar” case for k = 1, and was originally conceived for a quite different purpose, viz. completeness of eigenfunctions via elliptic partial differential equation theory.

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