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This paper investigates the discrete-time problem of the regularized Schrödinger 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.

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