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On the Cauchy problem for the Hartree approximation in quantum dynamics

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 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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03738486
Contributor : Rémi Carles Connect in order to contact the contributor
Submitted on : Tuesday, July 26, 2022 - 11:42:46 AM
Last modification on : Wednesday, August 3, 2022 - 4:00:40 AM

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  • HAL Id : hal-03738486, version 1
  • ARXIV : 2207.13928

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Rémi Carles, Clotilde Fermanian Kammerer, Caroline Lasser. On the Cauchy problem for the Hartree approximation in quantum dynamics. 2022. ⟨hal-03738486⟩

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