  We consider the generalized time-dependent Schr\"odinger equation on the
half-axis and a broad family of finite-difference schemes with the discrete
transparent boundary conditions (TBCs) to solve it. We first rewrite the
discrete TBCs in a simplified form explicit in space step $h$. Next, for a
selected scheme of the family, we discover that the discrete convolution in
time in the discrete TBC does not depend on $h$ and, moreover, it coincides
with the corresponding convolution in the semi-discrete TBC rewritten
similarly. This allows us to prove the bound for the difference between the
kernels of the discrete convolutions in the discrete and semi-discrete TBCs
(for the first time). Numerical experiments on replacing the discrete TBC
convolutions by the semi-discrete one exhibit truly small absolute errors
though not relative ones in general. The suitable discretization in space of
the semi-discrete TBC for the higher-order Numerov scheme is also discussed.
