The transition from diffusion to blow-up for a nonlinear Schroedinger equation in dimension 1

We consider the time-dependent one-dimensional nonlinear Schroedinger equation with a pointwise singular potential. We prove that if the strength of the nonlinear term is small enough, then the solution is well defined for any time, regardless of the choice of initial data; in contrast, if the nonlinearity power is larger than a critical value, for some initial data a blow-up phenomenon occurs in finite time. In particular, if the system is initially prepared in the ground state of the linear part of the Hamiltonian, then we obtain an explicit condition on the parameters for the occurrence of the blow-up