Perturbation theory for a Sturm-Liouville problem with an interior singularity

Abstract

Abstract The perturbation theory of operators and forms is used to construct Sturm-Liouville differential operators for potentials with I/x, Pf(1/x) and, for ϵ →0+, 1/(x-iϵ) interior singularities. The norm resolvent convergence of approximating sequences of operators with smooth potentials is established and various qualitative and quantitative properties of their spectra are obtained. For an infinite interval, a comparison is made with the one-dimensional Coulomb problem possessing a potential 1/|x|.