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Sturm–liouville eigenvalue problems in which the squares of the eigenfunctions are linearly dependent
Author(s) -
Mahar T. J.,
Willner B. E.
Publication year - 1980
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160330406
Subject(s) - eigenfunction , eigenvalues and eigenvectors , mathematics , sturm–liouville theory , mathematical analysis , pure mathematics , mathematical physics , quantum mechanics , physics , boundary value problem
We consider the eigenvalue problem u ″ + λϕ = 0, u(0)=u(1) = 0, where ϕεin C [0, 1] is positive. It is well known that the eigenfunctions corresponding to distinct eigenvalues are linearly independent. It is shown in this paper that the squares of the eigenfunctions may be linearly dependent on nontrivial subintervals of [0,1]. This result has relevance in the variational analysis of eigenvalue problems.