Accurate basis set by the CIP method for the solutions of the Schrödinger equation

In this paper, we propose a basis set approach by the Constrained Interpolation Profile (CIP) method for the calculation of bound and continuum wave functions of the Schrödinger equation. This method uses a simple polynomial basis set that is easily extendable to any desired higher-order accuracy. The interpolating profile is chosen so that the subgrid scale solution approaches the local real solution by the constraints from the spatial derivative of the original equation. Thus the solution even on the subgrid scale becomes consistent with the master equation. By increasing the order of the polynomial, this solution quickly converges. The method is tested on the one-dimensional Schrödinger equation and is proven to give solutions a few orders of magnitude higher in accuracy than conventional methods for the lower-lying eigenstates. The method is straightforwardly applicable to various types of partial differential equations.