Geodesic
Evaluation of eigenvalues and eigenfunctions of Coulomb spheroidal wave equation
Matematičeskoe modelirovanie, Tome 27 (2015) no. 7, pp. 111-116

Voir la notice de l'article provenant de la source Math-Net.Ru

A method for evaluation the eigenvalues $\lambda_{m,q}(b,c)$ and the eigenfunctions of Coulomb spheroidal wave equation in a case of complex parameters $b$ and $c$ is proposed. The method is based on construction of two expansions of solution at the singular points $\eta=\pm1$ and on matching of the expansions at the point $\eta=0$. Numerical experiments show that there exist branching points of the eigenvalues $\lambda_{m,q}(b,c)$ for some complex values $b$ or $c$, the order of the branching points is equal $2$.
Keywords: Coulomb spheroidal wave functions, complex parameters, evaluation of eigenvalues, branching points. 
S. L. Skorokhodov. Evaluation of eigenvalues and eigenfunctions of Coulomb spheroidal wave equation. Matematičeskoe modelirovanie, Tome 27 (2015) no. 7, pp. 111-116. http://geodesic.mathdoc.fr/item/MM_2015_27_7_a16/
@article{MM_2015_27_7_a16,
     author = {S. L. Skorokhodov},
     title = {Evaluation of eigenvalues and eigenfunctions of {Coulomb} spheroidal wave equation},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {111--116},
     year = {2015},
     volume = {27},
     number = {7},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2015_27_7_a16/}
}
TY  - JOUR
AU  - S. L. Skorokhodov
TI  - Evaluation of eigenvalues and eigenfunctions of Coulomb spheroidal wave equation
JO  - Matematičeskoe modelirovanie
PY  - 2015
SP  - 111
EP  - 116
VL  - 27
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/MM_2015_27_7_a16/
LA  - ru
ID  - MM_2015_27_7_a16
ER  - 
%0 Journal Article
%A S. L. Skorokhodov
%T Evaluation of eigenvalues and eigenfunctions of Coulomb spheroidal wave equation
%J Matematičeskoe modelirovanie
%D 2015
%P 111-116
%V 27
%N 7
%U http://geodesic.mathdoc.fr/item/MM_2015_27_7_a16/
%G ru
%F MM_2015_27_7_a16

[1] I. V. Komarov, L. I. Ponomarev, S. Iu. Slavianov, Sferoidalnye i kulonovskie sferoidalnye funktsii, Nauka, M., 1976

[2] D. B. Khrebtukov, “The exact numerical solution of a Schrodinger equation with two-Coulomb centers plus oscillator potential”, J. Physics A: Mathem. and General, 25 (1992), 3319–3328 | Zbl

[3] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington D. C., 1964

[4] L.-W. Li, X.-K. Kang, M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory, Wiley, N.-Y., 2001

[5] P. E. Falloon, P. C. Abbott, J. B. Wang, “Theory and Computation of the Spheroidal Wave Functions”, J. Physics A: Mathem. and General, 36 (2003), 5477–5495 | Zbl

[6] B. E. Barrowes, K. O'Neill, T. M. Grzegorczyk, J. A. Kong, “On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter”, Studies in Applied Mathematics, 113:3 (2004), 271–301 | Zbl

[7] E. L. Ince, Ordinary Differential Equations, Dover Publications, N.-Y., 1926

[8] S. L. Skorokhodov, D. V. Khristoforov, “Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions”, Computational Mathematics and Mathematical Physics, 46:7 (2006), 1132–1146

[9] S. L. Skorokhodov, D. V. Khristoforov, “Calculation of the branch points of the eigenvalues of the Coulomb spheroidal wave equation”, Computational Mathematics and Mathematical Physics, 47:11 (2007), 1802–1818

[10] A. A. Abramov, S. V. Kurochkin, “Highly accurate calculation of angular spheroidal functions”, Computational Mathematics and Mathematical Physics, 46:1 (2006), 10–15 | Zbl

[11] G. Blanch, D. S. Clemm, “The double points of Mathieu's differential equation”, Math. Comp., 23:105 (1969), 97–108 | Zbl

[12] E. A. Solov'ev, “The advanced adiabatic approach and inelastic transitions via hidden crossings”, J. Physics B: Atomic, Molecular and Optical Physics, 38 (2005), R153–R194