On the Cauchy problem for the Hartree approximation in quantum dynamics

Rémi Carles (1) , Clotilde Fermanian Kammerer (2) , Caroline Lasser (3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LAMA - Laboratoire Analyse et de Mathématiques Appliquées
3 TUM - Zentrum Mathematik [Munchen]

We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The Hartree equations of motion form a coupled system of nonlinear Schrödinger equations for the evolution of product state approximations. They are a prominent example for dimension reduction in the context of the the time-dependent Dirac-Frenkel variational principle. We handle the case of Coulomb potentials thanks to Strichartz estimates. Our main result addresses a general setting where the nonlinear coupling cannot be considered as a perturbation. The proof uses a recursive construction that is inspired by the standard approach for the Cauchy problem associated to symmetric quasilinear hyperbolic equations.

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

Rémi Carles, Clotilde Fermanian Kammerer, Caroline Lasser. On the Cauchy problem for the Hartree approximation in quantum dynamics. Nonlinearity, 2023, 36 (6), pp.3158-3181. ⟨10.1088/1361-6544/accf5a⟩. ⟨hal-03738486⟩