A trigonometrically-fitted one-step method with multi-derivative for the numerical solution to the one-dimensional Schrödinger equation

In this paper we present a new multi-derivative or Obrechkoff one-step method for the numerical solution to an one-dimensional Schrödinger equation. By using trigonometrically-fitting method (TFM), we overcome the traditional Obrechkoff one-step method (or called as the non-TFM) for its poor-accuracy in the resonant state. In order to demonstrate the excellent performance for the resonant state, we consider only the simplest TFM, of which the local truncation error (LTE) is of O(h⁷), a little higher than the one of the traditional Numerov method of O(h⁶), and only the first- and second-order derivatives of the potential function are needed. In the new method, in order to solve two unknowns, wave function and its first-order derivative, we use a pair of two symmetrically linear-independent one-step difference equations. By applying it to the well-known Woods-Saxon's potential problem, we find that the TFM can surpass the non-TFM by five orders for the highest resonant state, and surpass Numerov method by eight orders. On the other hand, because of the small error constant, the accuracy improvement to the ground state is also remarkable, and the numerical result obtained by TFM can be four to five orders higher than the one by Numerov method.