Estimating spectral density functions for Sturm–Liouville problems with two singular endpoints

In this paper, we consider the Sturm–Liouville equation −y′′ + qy = λy on the half line (0, ∞) under the assumptions that x=0 is a regular singular point and nonoscillatory for all real λ, and that either (i) q is L1 near x=∞, or (ii) q′ is L1 near ∞ with q(x) → 0 as x → ∞, so that there is absolutely continuous spectrum in (0, ∞). 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, ∞). 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 λ from 0 to 10000, accuracies which could not be achieved using the SLEDGE software package.