Spectral Method for Solving the Time-Dependent Schrödinger Equation on a Non-Uniform Coordinate Grid

The fundamental possibility of a numerical solution of the time-dependent Schrödinger equation for a wave function defined on a non-uniform coordinate grid by the spectral method using the fast Fourier transform algorithm is discussed. The method is based on reducing the non-uniform coordinate grid to a uniform one by a non-linear transformation of coordinates and approximating the obtained evolution operator using the Lie–Trotter–Suzuki product formula. Algorithms for the numerical solution of the first and second orders are developed.