New modified Runge–Kutta–Nyström methods for the numerical integration of the Schrödinger equation

In this work we construct new Runge–Kutta–Nyström methods especially designed to integrate exactly the test equation y″=−w2yy″=−w2y. We modify two existing methods: the Runge–Kutta–Nyström methods of fifth and sixth order. We apply the new methods to the computation of the eigenvalues of the Schrödinger equation with different potentials such as the harmonic oscillator, the doubly anharmonic oscillator and the exponential potential.