Project Team Ipso

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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 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 (S 2 ) with s>0, and not uniformly well-posed on L 2 (S 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 Schrödinger-Poisson system, singularly perturbed by an external potential that confines the particles in the vicinity of the sphere.