Limits of regularizations for generalized function solutions to the Schr\"odinger equation with "square root of delta" initial value

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.