  We deal with an initial-boundary value problem for the generalized
time-dependent Schr\"odinger equation with variable coefficients in an
unbounded $n$--dimensional parallelepiped ($n\geq 1$). To solve it, the
Crank-Nicolson in time and the polylinear finite element in space method with
the discrete transpa\-rent boundary conditions is considered. We present its
stability properties and derive new error estimates $O(\tau^2+|h|^2)$ uniformly
in time in $L^2$ space norm, for $n\geq 1$, and mesh $H^1$ space norm, for
$1\leq n\leq 3$ (a superconvergence result), under the Sobolev-type assumptions
on the initial function. Such estimates are proved for methods with the
discrete TBCs for the first time.
