Spectral properties of singular Sturm–Liouville operators via boundary triples and perturbation theory

Abstract

We apply both the theory of boundary triples and perturbation theory to the setting of semi-bounded Sturm–Liouville operators with two limit-circle endpoints. For general boundary conditions we obtain refined and new results about their eigenvalues and eigenfunctions.In the boundary triple setup, we obtain simple criteria for identifying which self-adjoint extensions possess double eigenvalues when the parameter is a matrix. We also identify further spectral properties of the Friedrichs extension and (when the operator is positive) the von Neumann–Krein extension.Motivated by some recent scalar Aronszajn–Donoghue type results, we find that real numbers can only be eigenvalues for two extensions of Sturm–Liouville operator when the boundary conditions are restricted to corresponding to affine lines in the space from which the perturbation parameter is taken. Furthermore, we determine much of the spectral representation of those Sturm–Liouville operators that can be reached by perturbation theory.