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 Zh. Vychisl. Mat. Mat. Fiz., 2007, Volume 47, Number 11, Pages 1880–1897 (Mi zvmmf222)

Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation

S. L. Skorokhodova, D. V. Khristoforovb

a Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119991, Russia
b Faculty of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, 119992, Russia

Abstract: A method for computing the eigenvalues $\lambda_{mn}(b,c)$ and the eigenfunctions of the Coulomb spheroidal wave equation is proposed in the case of complex parameters $b$ and $c$. The solution is represented as a combination of power series expansions that are then matched at a single point. An extensive numerical analysis shows that certain $b_s$ and $c_s$ are second-order branch points for $\lambda_{mn}(b,c)$ with different indices $n_1$ and $n_2$, so that the eigenvalues at these points are double. Padé approximants, quadratic Hermite–Padé approximants, the finite element method, and the generalized Newton method are used to compute the branch points $b_s$ and $c_s$ and the double eigenvalues to high accuracy. A large number of these singular points are calculated.

Key words: Coulomb spheroidal wave functions, computation of eigenvalues, branch point of eigenvalues, Padé approximants, quadratic approximations, generalized Newton method.

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English version:
Computational Mathematics and Mathematical Physics, 2007, 47:11, 1802–1818

Bibliographic databases:

UDC: 519.651
Revised: 23.05.2007

Citation: S. L. Skorokhodov, D. V. Khristoforov, “Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation”, Zh. Vychisl. Mat. Mat. Fiz., 47:11 (2007), 1880–1897; Comput. Math. Math. Phys., 47:11 (2007), 1802–1818

Citation in format AMSBIB
\Bibitem{SkoKhr07} \by S.~L.~Skorokhodov, D.~V.~Khristoforov \paper Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2007 \vol 47 \issue 11 \pages 1880--1897 \mathnet{http://mi.mathnet.ru/zvmmf222} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2405032} \elib{https://elibrary.ru/item.asp?id=13536202} \transl \jour Comput. Math. Math. Phys. \yr 2007 \vol 47 \issue 11 \pages 1802--1818 \crossref{https://doi.org/10.1134/S0965542507110073} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-36448992660} 

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This publication is cited in the following articles:
1. S. L. Skorokhodov, “Vychislenie sobstvennykh znachenii i sobstvennykh funktsii kulonovskogo volnovogo sferoidalnogo uravneniya”, Matem. modelirovanie, 27:7 (2015), 111–116
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