The Schr\"odinger-Poisson system on the sphere

We study the Schr\"odinger-Poisson system on the unit sphere $\SS^2$ of $\RR^3$, modeling the quantum transport of charged particles confined on a sphere by an external potential. Our first results concern the Cauchy problem for this system. We prove that this problem is regularly well-posed on every $H^s(\SS ^2)$ with $s>0$, and not uniformly well-posed on $L^2(\SS ^2)$. The proof of well-posedness relies on multilinear Strichartz estimates, the proof of ill-posedness relies on the construction of a counterexample which concentrates exponentially on a closed geodesic. In a second part of the paper, we prove that this model can be obtained as the limit of the three dimensional Schr\"odinger-Poisson system, singularly perturbed by an external potential that confines the particles in the vicinity of the sphere.