Titchmarsh–Weyl m ‐functions for second‐order Sturm–Liouville problems with two singular endpoints

Abstract

AbstractIn 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 eigenfunction expansion can be constructed. The use of initial conditions at a point interior to the interval (0,∞) is avoided in favor of Frobenius solutions near the regular singular point x = 0. In contrast to the classical theory associated with a regular left endpoint, the growth behaviour of the associated spectral functions can be on the order of λβ for any β ∈ (0,∞). Application of the theory to the Bessel equation on (0,∞) and to the radial part of the separated hydrogen atom on (0,∞) is given. In the case of the hydrogen atom a single Titchmarsh–Weyl m ‐function is obtained which completely describes both the discrete negative spectrum and the continuous positive spectrum. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)