  In this paper, we investigate the uniform exponential stability of a
semi-discrete scheme for a Schr\"{o}dinger equation under boundary feedback
stabilizing control in the natural state space $L^2(0,1)$. This study is
significant since a time domain energy multiplier that allows proving the
exponential stability of this continuous Schr\"{o}dinger system has not yet
found, thus leading to a major mathematical challenge to semi-discretization of
the PDE, an open problem for a long time. Although the powerful frequency
domain energy multiplier approach has been used in proving exponential
stability for PDEs since 1980s, its use to the \emph{uniform} exponential
stability of the semi-discrete scheme for PDEs has not been reported yet. The
difficulty associated with the uniformity is that due to the parameter of the
step size, it involves a family of operators in different state spaces that
need to be considered simultaneously. Based on the Huang-Pr\"{u}ss frequency
domain criterion for uniform exponential stability of a family of
$C_0$-semigroups in Hilbert spaces, we solve this problem for the first time by
proving the uniform boundedness for all the resolvents of these operators on
the imaginary axis. The proof almost exactly follows the procedure for the
exponential stability of the continuous counterpart, highlighting
  the advantage of this discretization method.
