Arbitrarily precise numerical solutions of the one-dimensional Schrödinger equation

In this paper, how to overcome the barrier for a finite difference method to obtain the numerical solutions of a one-dimensional Schrödinger equation defined on the infinite integration interval accurate than the computer precision is discussed. Five numerical examples of solutions with the error less than 10−50 and 10−30 for the bound and resonant state, respectively, obtained by the Obrechkoff one-step method implemented in the multi precision mode, which include the harmonic oscillator, the Pöschl–Teller potential, the Morse potential and the Woods–Saxon potential, demonstrate that the finite difference method can yield the eigenvalues of a complex potential with an arbitrarily desired precision within a reasonable efficiency.