On the global well-posedness of the quadratic NLS on $L^2(\mathbb{R}) + H^1(\mathbb{T})$

We study the one dimensional nonlinear Schr\"odinger equation with power nonlinearity $|u|^{\alpha - 1} u$ for $\alpha \in [1,5]$ and initial data $u_0 \in L^2(\mathbb{R}) + H^1(\mathbb{T})$. We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity ($\alpha = 2$) we obtain global well-posedness in the space $C(\mathbb{R}, L^2(\mathbb R) + H^1(\mathbb T))$ via Gronwall's inequality.