The semiclassical limit of the time dependent Hartree–Fock equation: The Weyl symbol of the solution

Amour, Laurent; Khodja, Mohamed R.; Nourrigat, Jean

doi:10.2140/apde.2013.6.1649

Cited by 36 publications

(101 citation statements)

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“…Moreover, the second result (Theorem 2.1) is a continuation of our work in [1], where the Weyl symbol was considered instead of the Wick symbol. The work in [1] is devoted to the the Weyl symbol of a solution ρ h (t) to the TDHF equation, under the hypothesis that ρ h (0) is a PDO trace class operator belonging to the class studied by Rondeaux [14].…”

Section: Introductionmentioning

confidence: 79%

“…Since we are concerned with trace class operators it makes sense that the norm involve to estimate the difference u h (., t)−v h (., t) is the L 1 (IR 2n ) norm. A first answer to this issue is given in [1] where we assume that the operator ρ h (0) is a PDO operator belonging to the Rondeaux class [14] (with a modification to take into account the parameter h). The article [1] also gives an asymptotic expansion at any order of u h (., t)−v h (., t).…”

Section: Statement Of the Second Result: The Case Of Trace Class Opermentioning

confidence: 99%

“…A first answer to this issue is given in [1] where we assume that the operator ρ h (0) is a PDO operator belonging to the Rondeaux class [14] (with a modification to take into account the parameter h). The article [1] also gives an asymptotic expansion at any order of u h (., t)−v h (., t). We consider here this problem with a weaker hypothesis than the one in [1].…”

Section: Statement Of the Second Result: The Case Of Trace Class Opermentioning

confidence: 99%

“…The work in [1] is devoted to the the Weyl symbol of a solution ρ h (t) to the TDHF equation, under the hypothesis that ρ h (0) is a PDO trace class operator belonging to the class studied by Rondeaux [14]. The motivation of the study of the Wick symbol rather than the Weyl symbol is the following.…”

Section: Introductionmentioning

confidence: 99%

“…The space L 1 (IR 2n ) is the natural function space for kinetic equations such as the Vlasov equation. However, the Weyl symbol of a trace class operator is not necessarily in L 1 (IR 2n ), without the strong hypotheses made in [1]. To the contrary, the Wick symbol of a trace class operator always belongs to L 1 (IR 2n ), and the use of the Wick symbol allows us to take very weak hypotheses.…”

Section: Introductionmentioning

confidence: 99%

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The classical limit of the Heisenberg and time-dependent Hartree–Fock equations: the Wick symbol of the solution

Amour

Khodja

Nourrigat

2013

Mathematical Research Letters

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Abstract. This paper is concerned with the Wick symbol of time evolving quantum observables. The time dynamics is following either the Heisenberg equation relative to the Schrödinger Hamiltonian, or the time-dependent Hartree-Fock equation. Under very weak assumptions, we prove that the Wick symbol approximatively follows the classical mechanics laws when the semiclassical parameter h tends to zero. For the Heisenberg equation, this is a form of what is commonly called the Ehrenfest theorem. These statements have to be understood in a weaker sense than usual and in return, we do not assume that the Weyl symbol of the initial observable belongs to a class allowing the use of the Egorov theorem. IntroductionThe aim of this paper is twofold: to prove, under minimal hypotheses, a version of Ehrenfest theorem, and to give a similar result for the time-dependent HartreeFock equation (TDHF). We first consider a quantum observable (bounded operator) A h (t), (t ∈ IR, h > 0), evolving according to the Heisenberg equation associated to the Schrödinger Hamiltonian (see (1.2)). Generally speaking, the Ehrenfest theorem suggests that the average of this observable A h (t), taken on an h-dependent coherent state, approximatively follows, for small h > 0, the classical mechanics laws [7]. The average of an observable on coherent states is related to its Wick symbol. The first goal of this work is to obtain minimal hypotheses on the initial data A h (0), in order that the idea conveyed by Ehrenfest is asymptotically verified when h tends to 0.A precise statement of Ehrenfest theorem, using Egorov theorem for the proof, is given in [4,13] (see also the bibliography therein). It is also noted ([4] and references therein) that the approximation by the classical mechanics laws is not valid uniformly for all time, but up to a maximal time called the Ehrenfest time. These results are obtained assuming that the family of observables (A h (0)) is a semiclassical pseudodifferential (PDO) operator (see [13] or [17]). Our first goal is therefore to give another precise statement of Ehrenfest theorem, weaker than the one in [13], but in assuming weaker hypotheses on A h (0) (Theorem 1.1). In particular, we do not assume that A h (0) is a PDO operator. It is known that an operator in L 2 (IR n ) is a PDO one in the standard class of Calderón and Vaillancourt, when all its iterated commutators with position and momentum operators are bounded in L 2 (IR n ) (Beals characterization theorem [2]). In our work, the main assumption is that the single commutators of A h (0) with position and momentum operators are bounded, without any hypotheses on the iterated commutators. Then we prove that, in some sense, the

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Section: Introductionmentioning

confidence: 79%

Section: Statement Of the Second Result: The Case Of Trace Class Opermentioning

confidence: 99%

Section: Statement Of the Second Result: The Case Of Trace Class Opermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The classical limit of the Heisenberg and time-dependent Hartree–Fock equations: the Wick symbol of the solution

Amour

Khodja

Nourrigat

2013

Mathematical Research Letters

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Mean-Field Evolution of Fermions with Singular Interaction

Saffirio

2018

Springer Proceedings in Mathematics &Amp; Statistics

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We consider a system of N fermions in the mean-field regime interacting though an inverse power law potential V (x) = |x| −α , for α ∈ (0, 1]. We prove the convergence of a solution of the many-body Schrödinger equation to a solution of the time-dependent Hartree-Fock equation in the sense of reduced density matrices. We stress the dependence on the singularity of the potential in the regularity of the initial data. The proof is an adaptation of [22], where the case α = 1 is treated.

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From the Hartree to the Vlasov Dynamics: Conditional Strong Convergence

Saffirio

2021

Springer Proceedings in Mathematics &Amp; Statistics

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The semiclassical limit of the time dependent Hartree–Fock equation: The Weyl symbol of the solution

Cited by 36 publications

References 26 publications

The classical limit of the Heisenberg and time-dependent Hartree–Fock equations: the Wick symbol of the solution

The classical limit of the Heisenberg and time-dependent Hartree–Fock equations: the Wick symbol of the solution

Mean-Field Evolution of Fermions with Singular Interaction

From the Hartree to the Vlasov Dynamics: Conditional Strong Convergence

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