On the Dirac delta as initial condition for nonlinear Schr\"odinger equations

In this article we will study the initial value problem for some Schr"odinger equations with Diraclike initial data and therefore with infinite L2 mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This allows us to conclude a stability result in the defocusing setting. These problems are related to the existence of a singular dynamics for Schr"odinger maps through the so called Hasimoto transformation.