An accurate spectral method with arbitrarily large time step sizes

We propose a new implementation of the spectral method based on the Crank-Nicholson scheme. Our approach introduces an effective system Hamiltonian that shares the same eigenstates with the original system. The eigenenergies of the effective system Hamiltonian are related to those of the original system in a simple analytical form. The exact and unconditionally convergent propagator for the effective system allows easy and accurate recovery of the high resolution energy spectra of the original system with arbitrarily large time step sizes. Numerical results are given for a Morse oscillator.