Estimating Spectral Density Functions for Sturm-Liouville problems with two singular endpoints

TL;DR

This paper develops methods to accurately estimate spectral density functions for Sturm-Liouville problems with two singular endpoints, enabling high-precision numerical results for complex quantum systems.

Contribution

It introduces new algorithms and characterizations for spectral density functions in doubly singular Sturm-Liouville problems, improving numerical accuracy over existing software.

Findings

Achieves near machine precision accuracy for spectral density functions.

Successfully applies algorithms to the radial hydrogen atom problem.

Outperforms existing software like SLEDGE in accuracy.

Abstract

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 [12] and [13] (when the left endpoint is regular) are established; corresponding approximants from the two algorithms in [12] and [13] are then utilized, along with the 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 0.1 to 10000, accuracies which could not be achieved using the SLEDGE software package.