Smoothing effect and exponential stability of discrete-time Schrodinger equation with fractional regularization term

This paper investigates the discrete-time problem of the regularized Schrodinger equation with fractional Laplacian term in bounded domains. We proved the exponential stability and the Kato smoothing effect of the discrete-time Crank-Nicolson scheme. The proofs are based on uniform estimates of the semi-discrete resolvent obtained by the Z-Transformation. We constructed a well-posed approximation scheme and carried out numerical simulations to verify numerically the exponential stability and the smoothing effect.