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arXiv:1202.1065 · Journal of Mathematical Physics

Mean-field quantum dynamics with magnetic fields

Jonas Lührmann

Published Feb 1, 2012• 18 citations
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Abstract

We consider a system of N bosons in three dimensions interacting through a mean-field Coulomb potential in an external magnetic field. For initially factorized states we show that the one-particle density matrix associated with the solution of the N-body Schrödinger equation converges to the projection onto the solution of the magnetic Hartree equation in trace norm and in energy as N → ∞. Estimates on the rate of convergence are provided.

DOI
10.1063/1.3687024

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We consider a system of N bosons in three dimensions interacting through a mean-field Coulomb potential in an external magnetic field. For initially factorized states we show that the one-particle density matrix associated with the solution of the N-body Schrödinger equation converges to the projection onto the solution of the magnetic Hartree equation in trace norm and in energy as N → ∞. Estimates on the rate of convergence are provided.
We propose a new approach for the study of the time evolution of a factorized N-particle bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the many-particle Schrödinger dynamics and the one-particle nonlinear Hartree dynamics. In particular the one-particle density matrix associated with the solution to the N-particle Schrödinger equation is shown to converge to the projection onto the one-dimensional subspace spanned by the solution to the Hartree equation with a speed of convergence of order 1/N for all fixed times.
We study the quantum dynamics of a large number of interacting fermionic particles in a constant magnetic field. In a coupled mean-field and semiclassical scaling limit, we show that solutions of the many-body Schrödinger equation converge to solutions of a non-linear Hartree-Fock equation. The central ingredient of the proof are certain semiclassical trace norm estimates of commutators of the position and momentum operators with the one-particle density matrix of the solution of the Hartree-Fock equation. In a first step, we prove their validity for non-interacting initial data in a magnetic field by generalizing a 2020 result of Fournais and Mikkelsen. We then propagate these bounds from the initial data along the Hartree-Fock flow to arbitrary times.
The derivation of the Hartree equation from many-body systems of Bosons in the mean field limit has been very intensively studied in the last couple of years. However, very few results exist showing convergence of the kth marginal of the N-body density matrix to the projection to the k-fold tensor product of the solution of the Hartree equation in stronger trace norms like the energy trace norm, see the work of Michelangeli and Schlein [Commun. Math. Phys. 311(3), 645–687 (2012)] and Lührmann [J. Math. Phys. 53(2), 022105 (2012)]. This issue is from a physical view point very important. The reason is that one can then approximate expectation values of certain observables of the N-body system by means of the Hartree equation, with relaxation of the very restrictive assumption that the observables are bounded operators. Here we consider the non-relativistic case. We prove, assuming only H1-regularity of the initial data, convergence in the energy trace norm without rates, and convergence in any other weaker Sobolev trace norm with rates. Our proof is simple and uses the functional aN introduced by Pickl [Lett. Math. Phys. 97(2), 151–164 (2011)].
We consider the dynamics of N interacting Bosons in three dimensions which are strongly confined in one or two directions. We analyze the mean field scaling and the nonlinear Schrödinger scaling of the interaction potential w and choose the initial wavefunction to be close to a product wavefunction. For both scalings we prove that in the mean field limit the dynamics of the N-particle system is described by a nonlinear equation in two or one dimensions. In the case of the mean field scaling this equation is the Hartree equation and for the second scaling the nonlinear Schrödinger equation. In both cases we obtain explicit bounds for the rate of convergence of the N-particle dynamics to the one-particle dynamics.
We consider the dynamics of a large number [Formula: see text] of nonrelativistic bosons in the mean field limit for a class of interaction potentials that includes Coulomb interaction. In order to describe the fluctuations around the mean field Hartree state, we introduce an auxiliary Hamiltonian on the [Formula: see text]-particle space that is similar to the one obtained from Bogoliubov theory. We show convergence of the auxiliary time evolution to the fully interacting dynamics in the norm of the [Formula: see text]-particle space. This result allows us to prove several other results: convergence of reduced density matrices in trace norm with optimal rate, convergence in energy trace norm, and convergence to a time evolution obtained from the Bogoliubov Hamiltonian on Fock space with expected optimal rate. We thus extend and quantify several previous results, e.g., by providing the physically important convergence rates, including time-dependent external fields and singular interactions, and allowing for more general initial states, e.g., those that are expected to be ground states of interacting systems.