Time-Splitting Chebyshev-Spectral Method for the Schrödinger Equation in the Semiclassical Regime with Zero Far-Field Boundary Conditions

Semiclassical limit of Schrödinger equation with zero far-field boundary conditions is investigated by the time-splitting Chebyshev-spectral method. The numerical results of the position density and current density are presented. The time-splitting Chebyshev-spectral method is based on Strang splitting method in time coupled with Chebyshev-spectral approximation in space. Compared with the time-splitting Fourier-spectral method, the time-splitting Chebyshev-spectral method is unnecessary to treat the wave function as periodic and holds the smoothness of the wave function. For different initial conditions and potential (e.g. constant potential and harmonic potential), extensive numerical test examples in one-dimension are studied. The numerical results are in good agreement with the weak limit solutions. It shows that the time-splitting Chebyshev-spectral method is effective in capturing ?-oscillatory solutions of the Schrödinger equation with zero far-field boundary conditions. In addition, the time-splitting Chebyshev-spectral method surpasses the traditional finite difference method in the meshing strategy due to the exponentially high-order accuracy of Chebyshev-spectral method.