An efficient second-order finite difference method for the one-dimensional schrödinger equation with absorbing boundary conditions

A stable and convergent second-order fully discrete finite difference scheme with efficient approximation of the exact absorbing boundary conditions is proposed to solve the Cauchy problem of the one-dimensional Schrödinger equation. Our approximation is based on the Padé expansion of the square root function in the complex plane. By introducing a constant damping term to the governing equation and modifying the standard Crank–Nicolson implicit scheme, we show that the fully discrete numerical scheme is unconditionally stable if the order of Padé expansion is chosen from our criterion. In this case, an optimal-order asymptotic error estimate is proved for the numerical solutions. Numerical examples are provided to support the theoretical analysis and illustrate the performance of the proposed numerical scheme.