  We briefly review results on Colombeau type generalized solutions to the
Cauchy problem for linear Schr\"odinger-type equations with non-smooth
principal part and their compatibility with classical and distributional
solutions. In the main part, we study convergence properties of regularized
solutions to the standard Schr\"odinger equation with initial values
corresponding to "square roots" of Dirac measures in various duals of classical
subspaces of the space of continuous functions.
