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arXiv · 2207.13928

On the Cauchy problem for the Hartree approximation in quantum dynamics

Rémi Carles·Clotilde Fermanian Kammerer·Caroline Lasser
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Abstract

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 Schr{\"o}dinger 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.

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Rémi Carles, Clotilde Fermanian Kammerer, Caroline Lasser. 2022-07-28. On the Cauchy problem for the Hartree approximation in quantum dynamics. https://doi.org/10.1088/1361-6544%2Faccf5a

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