Variationally Improved Spectral Method as an extremely accurate technique for solving time-independent Schrödinger equation

We introduce three distinct, yet equivalent, optimization procedures for the Fourier Spectral Method which increase its accuracy. This optimization procedure also allows us to uniquely define the error for the cases which are not exactly solvable, and this error matches closely its counterpart for the cases which are exactly solvable. Moreover, this method is very simple to program, fast, extremely accurate (e.g. an error of order 10−130 is usually obtainable as compared to the exact results), very robust and stable. Most importantly, one can obtain the energies and the wave functions of as many of the bound states as desired with a single run of the algorithm. We first thoroughly test this method against an exactly solvable problem and then apply it to two problems which are not exactly solvable, all for finding the bound states of the time-independent Schrödinger equation. We present a detailed comparison between the results obtained by this method and some of the more routine methods.