We study the one dimensional nonlinear Schrödinger equation with power nonlinearity | u| ^α-1u for α∈ [1 , 5] and initial data u∈ H¹(T) + L²(R). We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity (α= 2) we obtain global well-posedness in the space C(R, H¹(T) + L²(R)) via Gronwall’s inequality.