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.