Sturm–liouville eigenvalue problems in which the squares of the eigenfunctions are linearly dependent

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. Copyright © 1980 Wiley Periodicals, Inc., A Wiley Company