Fast LP method for the Schrödinger equation

The LP and CP methods are two versions of the piecewise perturbation methods to solve the Schrödinger equation. On each step the potential function is approximated by a constant (for CP) or by a linear function (for LP) and the deviation of the true potential from this approximation is treated by the perturbation theory.This paper is based on the idea that an LP algorithm can be made faster if expressed in a CP-like form. We obtain a version of order 12 whose two main ingredients are a new set of formulae for the computation of the zeroth-order solution which replaces the use of the Airy functions, and a convenient way of expressing the formulae for the perturbation corrections. Tests on a set of eigenvalue problems with a very big number of eigenvalues show that the proposed algorithm competes very well with a CP version of the same order and is by one order of magnitude faster than the LP algorithms existing in the literature. We also formulate a new technique for the step width adjustment and bring some new elements for a better understanding of the energy dependence of the error for the piecewise perturbation methods.