Open AccessOn the multiplicity of the eigenvalues of the vectorial Sturm-Liouville equation
Author(s) -
Chien-Wen Lin
Publication year - 2011
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.42.2011.764
Subject(s) - multiplicity (mathematics) , mathematics , eigenvalues and eigenvectors , sturm–liouville theory , lambda , dirichlet distribution , algebraic number , pure mathematics , mathematical analysis , combinatorics , physics , boundary value problem , quantum mechanics
Let $ Q(x) $ be a continuous $ mimes m $ real symmetric matrix-valued function defined on $ [0,1] $, and denote the Sturm-Liouville operator $ -frac{d^2}{dx^2}+Q(x) $ as $ L_Q $ with $ Q(x)$ as its potential function. In this paper we prove that for each Dirichlet eigenvalue $ lambda_* $ of $L_Q$, the geometric multiplicity of $ lambda_* $ is equal to its algebraic multiplicity. Applying this result, we get a necessary and sufficiently condition such that each Dirichlet eigenvalue of $ L_Q $ is of multiplicity $ m $