Journal
IMA JOURNAL OF NUMERICAL ANALYSIS
Volume 34, Issue 2, Pages 609-650Publisher
OXFORD UNIV PRESS
DOI: 10.1093/imanum/drt045
Keywords
Sturm-Liouville problem; spectral density function; spectral function; initial value problem; regular singular point; Frobenius power series solution; Whittaker functions; asymptotic expansions; trigonometric spline
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Funding
- National Science Foundation [DMS-0109022]
- Engineering and Physical Sciences Research Council [S63403/01]
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In this paper, we consider the Sturm-Liouville equation -y'' + qy = lambda y on the half line (0, infinity) under the assumptions that x=0 is a regular singular point and nonoscillatory for all real lambda, and that either (i) q is L-1 near x=infinity, or (ii) q' is L-1 near infinity with q(x) -> 0 as x -> infinity, so that there is absolutely continuous spectrum in (0, infinity). Characterizations of the spectral density function for this doubly singular problem, similar to those obtained in Fulton et al. (2008, J. Comp. Appl. Math., 212, 194-213) and Fulton et al. (2008, J. Comp. Appl. Math., 212, 150-178) (when the left endpoint is regular) are established; corresponding approximants from the two algorithms in these papers are then utilized, along with Frobenius recurrence relations and piecewise trigonometric/hyperbolic splines, to generate numerical approximations to the spectral density function associated with the doubly singular problem on (0, infinity). In the case of the radial part of the separated hydrogen atom problem, the new algorithms are capable of achieving near machine precision accuracy over the range of lambda from 0 to 10000, accuracies which could not be achieved using the SLEDGE software package.
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