A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems

In this article we develop a family of three explicit symmetric linear four-step methods. The new methods, with nullified phase-lag, are optimized for the efficient solution of the Schrödinger equation and related oscillatory problems. 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, where we find that there is a direct relationship between the periodicity intervals of the methods and their local truncation errors. In addition we determine their periodicity regions. We finally 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.