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

✍ Scribed by Charles Fulton; David Pearson; Steven Pruess

Publisher
Elsevier Science
Year
2008
Tongue
English
Weight
244 KB
Volume
212
Category
Article
ISSN
0377-0427

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✦ Synopsis

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.

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## Abstract In this paper we consider some cases of Sturm–Liouville problems with two singular endpoints at __x__ = 0 and __x__ = __∞__ which have a simple spectrum, and show that the simplicity of the spectrum can be built into the definition of a Titchmarsh–Weyl __m__ ‐function from which the eig