The Schrödinger-Poisson System on the Sphere

Abstract. We study the Schrödinger–Poisson system on the unit sphere S2 of R3, 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 Hs(S2) with s > 0, and not uniformly well-posed on L2(S2). The proof of well-posedness relies on multilinear Strichartz estimates, and 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 Schrödinger–Poisson system, singularly perturbed by an external potential that confines the particles in the vicinity of the sphere.