New characterizations of spectral density functions for singular Sturm–Liouville problems

A generalization is given for a characterization of the spectral density function of Weyl and Titchmarsh for a singular Sturm–Liouville problem having absolutely continuous spectrum in [ 0 , ∞ ) . A recurrent formulation is derived that generates a family of approximations based on this scheme. Proofs of convergence for these new approximations are supplied and a numerical method is implemented. The computational results show more rapid rates of convergence which are in accord with the theoretical rates.