The radial Schrödinger equation for a spherically symmetric potential can be regarded as a one-dimensional classical harmonic oscillator with a time-dependent spring constant. For solving classical dynamics problems, symplectic integrators are well known for their excellent conservation properties. The class of gradient symplectic algorithms is particularly suited for solving harmonic-oscillator dynamics. By use of Suzuki’s rule [Proc. Jpn. Acad., Ser. B: Phys. Biol. Sci.69, 161 (1993)] for decomposing time-ordered operators, these algorithms can be easily applied to the Schrödinger equation. We demonstrate the power of this class of gradient algorithms by solving the spectrum of highly singular radial potentials using Killingbeck’s method [J. Phys. A18, 245 (1985)] of backward Newton-Ralphson iterations.

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