  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.
