consider the $(1+1)D$ or $(2+1)D$ NLS: $$ i\psi _t (t,{\bf x}) + \Delta \psi + |\psi|^2\psi = 0 \, ,$$ $$ \psi (t,|{\bf x}|\to \infty) =0 \, , \quad \psi(t=0,{\bf x}) = \psi_0 ({\bf x}) \, , $$ with $x\in \mathbb{R}^d$, $d=1,2$, $t\geq0$. Usually, the discussion about solutions or blow-up of the NLS is for $\psi_0 \in H^1$, and there is some work about "rougher" initial condition.
My question: is what sort of analysis is done for the case of $\psi_0 = \delta ({\bf x})$. Obviously, this can not be an initial condition in the strict sense, but I do wonder if some work was done in this direction?
I was referred to this work, but I will appreciate further references.
