Knocking out teeth in one-dimensional periodic NLS

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schr\"odinger equation in one dimension with initial data $u_{0}$ in $H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T), 0\leq s_{1}\leq s_{2}.$ In addition, we show that if $u_{0}\in H^{s}(\mathbb R)+H^{\frac12+\epsilon}(\mathbb T)$ where $\epsilon>0$ and $\frac16\leq s\leq\frac12$ the solution is unique in $H^{s}(\mathbb R)+H^{\frac12+\epsilon}(\mathbb T).$ Our main tool is a normal form type reduction via the use of the differentiation by parts technique.