A note on the basis set approach in the constrained interpolation profile method

We propose a simple polynomial basis set that is easily extendable to any desired higher-order accuracy. This method is based on the Constrained Interpolation Profile (CIP) method and the profile is chosen so that the subgrid scale solution approaches the real solution by the constraints from the spatial derivative of the original equation. By adopting the higher-order derivatives of the master equations as constraints to generate a self-consistent subgrid profile, this solution quickly converges. 3rd and 5th order polynomials are tested on the one-dimensional Schrِdinger equation and are proved to give solutions a few orders of magnitude higher in accuracy than conventional methods for lower-lying eigenstates.