On the Distribution of the Eigenvalues and the Order of the Eigenfunctions of a Fourth-order Singular Boundary Value Problem

This paper is concerned with the asymptotic properties of the eigenvalues and eigenfunctions of the boundary value problem [formula omitted] With suitable restrictions placed on the real-valued coefficient q the spectrum of this problem, with respect to the eigenvalue parameter λ, is discrete; let {λn; n = 1, 2,…} and {ψ; n = 1, 2,…} be the eigenvalues and associated eigenfunctions. Asymptotic formulae are obtained for N(λ), the number of eigenvalues not exceeding the real number λ, and for ψn.,(x) as n-→∞ where x is a fixed, positive real number. © 1972, Royal Society of Edinburgh. All rights reserved.