A Class of Nonlinear Schrِdinger Equations with Concentrated Nonlinearity

We consider the nonlinear Schrödinger equation in dimension one with a nonlinearity concentrated in a finite number of points. Detailed results on the local existence of the solution in fractional Sobolev spaces Hρ are given. We also prove the conservation of the L2-norm and the energy of the solution and give a global existence result for repulsive and weakly attractive interaction in the space H1. Finally we prove the existence of blow-up solutions for strongly attractive interaction.