Singular Dirac Systems and Sturm–Liouville Problems Nonlinear in the Spectral Parameter

Abstract

For the Dirac operator with spherically symmetric potential V: (0,∞)→R we investigate the problem of whether the boundary points of the essential spectrum are accumulation points of discrete eigenvalues or not. Our main result shows that the accumulation of such eigenvalues is essentially determined by the asymptotic behaviour of V at 0 and ∞. We obtain this result by using a Levinson-type theorem for asymptotically diagonal systems depending on some parameter, a comparison theorem for the principal solutions of singular Dirac systems, and some criteria on the eigenvalue accumulation (respectively, non-accumulation) of λ-nonlinear singular Sturm–Liouville problems.