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

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. -->

DOI:	10.48550/arxiv.1303.2989
Přístupová URL adresa:	https://explore.openaire.eu/search/publication?articleId=doi_dedup___::304ea156a48b16dde89bf549a6146f2b
Rights:	OPEN
Přírůstkové číslo:	edsair.doi.dedup.....304ea156a48b16dde89bf549a6146f2b
Autor:	Charles T. Fulton, David Pearson, Steven Pruess
Rok vydání:	2013
Předmět:	0209 industrial biotechnology Regular singular point Recurrence relation 65L15 34B20 34B24 34B30 Applied Mathematics General Mathematics media_common.quotation_subject 020208 electrical & electronic engineering Mathematical analysis Spectrum (functional analysis) Sturm–Liouville theory 02 engineering and technology Numerical Analysis (math.NA) Infinity Lambda Computational Mathematics 020901 industrial engineering & automation 0202 electrical engineering electronic engineering information engineering Piecewise FOS: Mathematics Point at infinity Mathematics - Numerical Analysis Mathematics media_common
DOI:	10.48550/arxiv.1303.2989
Popis:	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.
Databáze:	OpenAIRE
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