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Pseudospectral methods on a semi-infinite interval with application to the hydrogen atom: A comparison of the mapped Fourier-sine method with Laguerre series and rational Chebyshev expansions. (English) Zbl 1028.65086
Summary: The Fourier-sine-with-mapping pseudospectral algorithm of E. Fattal, R. Baer, and R. Kostoff [Phase space approach for optimizing grid representations: The mapped Fourier method. Phys. Rev. E 53, 1217 (1996)] has been applied in several quantum physics problems. Here, we compare it with pseudospectral methods using Laguerre functions and rational Chebyshev functions. We show that Laguerre and Chebyshev expansions are better suited for solving problems in the interval $r\in \left[0,\infty \right]$ (for example, the Coulomb–Schrödinger equation), than the Fourier-sine-mapping scheme. All three methods give similar accuracy for the hydrogen atom when the scaling parameter $L$ is optimum, but the Laguerre and Chebyshev methods are less sensitive to variations in $L$. We introduce a new variant of rational Chebyshev functions which has a more uniform spacing of grid points for large $r$, and gives somewhat better results than the rational Chebyshev functions of J. P. Boyd [J. Comput. Phys. 70, 63-88 (1987; Zbl 0614.42013)].
##### MSC:
 65L15 Eigenvalue problems for ODE (numerical methods)