2013/03/12 by Charles T. Fulton, Fulton, Charles, D. B. Pearson +4
Mathematics · Physics and Astronomy · #34B20 #34B24 #34B30 #65L15 #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical Analysis (math.NA) #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1303.2989
openalex publication_date 2013/03/12 · openalex created_date 2022/09/20 · openalex updated_date 2026/07/28
In this paper we consider the Sturm-Liouville equation -y"+qy = lambda*y on\nthe half line (0,infinity) under the assumptions that x=0 is a regular singular\npoint and nonoscillatory for all real lambda, and that either (i) q is L1 near\nx=infinity, or (ii) q' is L1 near infinity with q(x) --> 0 as x --> infinity,\nso that there is absolutely continuous spectrum in (0,infinity).\nCharacterizations of the spectral density function for this doubly singular\nproblem, similar to those obtained in [12] and [13] (when the left endpoint is\nregular) are established; corresponding approximants from the two algorithms in\n[12] and [13] are then utilized, along with the Frobenius recurrence relations\nand piecewise trigonometric - hyperbolic splines, to generate numerical\napproximations to the spectral density function associated with the doubly\nsingular problem on (0,infinity). In the case of the radial part of the\nseparated hydrogen atom problem, the new algorithms are capable of achieving\nnear machine precision accuracy over the range of lambda from 0.1 to 10000,\naccuracies which could not be achieved using the SLEDGE software package.\n