Well- and ill-posedness of the Cauchy problem for semi-linear Schrödinger equations on the torus

We consider the Cauchy problem for semi-linear Schrödinger equations on the torus $\mathbb{T}$. We establish a necessary and sufficient condition on the polynomial nonlinearity for the Cauchy problem to be well-posed in the Sobolev space $H^s(\mathbb{T})$ for $s>\frac{5}{2}$. For the well-posedness, we use the energy estimates and the gauge transformation. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem.