Section: New Results

The Schrödinger Poisson system on the sphere

Participant : Florian Méhats.

In [31] we study the Schrödinger-Poisson system on the unit sphere ${S}^{2}$ of ${ℝ}^{3}$, 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 ${H}^{s}\left({S}^{2}\right)$ with $s>0$, and not uniformly well-posed on ${L}^{2}\left({S}^{2}\right)$. 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 Schrödinger-Poisson system, singularly perturbed by an external potential that confines the particles in the vicinity of the sphere.