Abstract. We consider the nonlinear Schr\"{o}dinger 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 $H^{\rho}$ are given. We also prove the conservation of the $L^{2}$-norm and the energy of the solution and give a global existence result for repulsive and weakly attractive interaction in the space $H^{1}$. Finally we prove the existence of blow-up solutions for strongly attractive interaction.