We develop two new explicit Runge-Kutta-Nyström methods for the efficient numerical solution of the Schrödinger equation. The construction of the first one method is based on a Runge-Kutta-Nyström method of Dormand, El-Mikkawy and Prince ([1]). The second one is a phase-fitted and amplification-fitted explicit Runge-Kutta-Nyström method. The latter method has infinite order of phase-lag and amplification factor as compared to the corresponding new constructed method with constant coefficients and other methods from the literature.

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