Dependence of eigenvalues on the boundary conditions of Sturm–Liouville problems with one singular endpoint

Abstract

This paper is concerned with continuous and differentiable dependence of isolated eigenvalues on the boundary conditions of self-adjoint Sturm–Liouville problems with one singular endpoint. Locally continuous dependence of eigenvalues on the boundary conditions is proved. Especially, in the limit point case, the continuous dependence of isolated eigenvalues is shown by the relationships between Weyl–Titchmarsh m(λ)-function and the spectrum of the singular problems. Then continuous eigenvalue branches through each isolated eigenvalue over the space of boundary conditions are formed. It is rigorously shown that the eigenfunctions for the eigenvalues along a continuous simple eigenvalue branch can be continuously chosen with uniform bound in Lw2 norm. Its proof is different from that in the regular Sturm–Liouville problems. The derivative formulas of the continuous simple eigenvalue branch with respect to all the parameters in the boundary conditions are given, and thus its monotonicity with respect to some parameters is derived.