CP methods for the Schrِdinger equation revisited

On constructing CPM propagators with an abundant number of terms by MATHEMATICA, we have shown that the CPM[N, Q], where N is the number of polynomial terms by which the potential is approximated in each interval and Q the number of corrections introduced, is a method of order 2N + 2 at low energies if Q ⩾ ⌊23N⌋ + 1 and of order N at high energies if Q ⩾ 1. We have also proved that in the last case the error damps out as 1E for both initial- and boundary-value problems. We have written a program for boundary-value problems which is of order 12, 10 at low and high energies, respectively, and have found out that it is far more efficient than the well-established codes SL02F, SLEDGE and SLEIGN.