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Numer. Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>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 Schr\u00f6dinger equation. Our approximation is based on the Pad\u00e9 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 Pad\u00e9 expansion is chosen from our criterion. In this case, an optimal-order asymptotic error estimate is proved for the numerical solutions. 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