Titchmarsh–Weyl m -functions for second-order Sturm–Liouville problems with two singular endpoints
✍ Scribed by Charles Fulton
- Publisher
- John Wiley and Sons
- Year
- 2008
- Tongue
- English
- Weight
- 518 KB
- Volume
- 281
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
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 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 λ^β^ 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)