n this paper, we will present a new six-step P-stable method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the one-dimensional Schrödinger equation. We perform an analysis of the local truncation error of the methods for the general case and for the special case of the Schrödinger equation, where we show the decrease of the maximum power of the energy in relation to the corresponding classical methods. We also perform a periodicity analysis. In addition, we determine their periodicity regions. Finally, we compare the new methods to the corresponding classical ones and other known methods from the literature, where we show the high efficiency of the new methods.
