Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation

The numerical properties of a leap-frog pseudospectral scheme for the Schrödinger equation are analyzed. Stability, second-order accuracy in time, and spectral accuracy in space are discussed considering the linear Schrödinger equation with potential in a periodic setting. Further issues regarding phase error, gauge invariance, conservation properties, and commutation relations are addressed. Results of numerical experiments are reported to demonstrate the validity and limitations of the theoretical findings and for comparison with the well known Crank–Nicholson finite difference scheme.