Scattering Theory for the Schrödinger Equation in some external time dependent magnetic fields

J. Ginibre (1) , G. Velo

1 LPT - Laboratoire de Physique Théorique d'Orsay [Orsay]

We study the theory of scattering for a Schrödinger equation in an external time dependent magnetic field in the Coulomb gauge, in space dimension 3. The magnetic vector potential is assumed to satisfy decay properties in time that are typical of solutions of the free wave equation, and even in some cases to be actually a solution of that equation. That problem appears as an intermediate step in the theory of scattering for the Maxwell-Schrödinger (MS) system. We prove in particular the existence of wave operators and their asymptotic completeness in spaces of relatively low regularity. We also prove their existence or at least results going in that direction in spaces of higher regularity. The latter results are relevant for the MS system. As a preliminary step, we study the Cauchy problem for the original equation by energy methods, using as far as possible time derivatives instead of spaces derivatives.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique mathématique [math-ph]; Algèbres quantiques [math.QA]

J. Ginibre, G. Velo. Scattering Theory for the Schrödinger Equation in some external time dependent magnetic fields. Journal of Differential Equations, 2005, 215, pp.108-177. ⟨hal-00013677⟩