Laurent Lafleche
In this paper, we prove a quantitative version of the semiclassical limit from the Hartree to the Vlasov equation with singular interaction, including the Coulomb potential. To reach this objective, we also prove the propagation of velocity moments and weighted Schatten norms which implies the boundedness of the space density of particles uniformly in the Planck constant.
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Strong semiclassical limit from Hartree and Hartree-Fock to Vlasov-Poisson equation
本文研究了从Hartree和Hartree-Fock到Vlasov方程的半经典极限，在Schatten范数下证明了强拓扑中的显式界，并给出了费米子情形下交换项估计及收敛速率，适用于一般初始数据和任意固定时间区间。
We review the recent results [45, 46] concerning the semiclassical limit from the Hartree dynamics to the Vlasov equation with singular potentials and extend them to the case of more general radial interactions. We prove that, at positive temperature, the Hartree dynamics converges in trace norm to the Vlasov one, for a particular class of initial states.
We consider the nonlinear Hartree and Vlasov equations around a translation-invariant (homogeneous) stationary state in infinite volume, for a short range interaction potential. For both models, we consider time-dependent solutions which have a finite relative energy with respect to the reference translation-invariant state. We prove the convergence of the Hartree solutions to the Vlasov ones in a semi-classical limit and obtain as a by-product global well-posedness of the Vlasov equation in the (relative) energy space.
We consider the evolution of $N$ fermions interacting through a Coulomb or gravitational potential in the mean-field limit as governed by the nonlinear Hartree equation with Coulomb or gravitational interaction. In the limit of large $N$, we study the convergence in trace norm towards the classical Vlasov-Poisson equation for a special class of mixed quasi-free states.
Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data
本文在维度 d≥3 的奇异势（含三维库仑势）情形下，针对集中初值证明了从Hartree方程到Vlasov方程的定量全局半经典极限，并给出空间与速度矩的半经典界及空间密度渐近行为。
On the $L^2$ Rate of Convergence in the Limit from the Hartree to the Vlasov$\unicode{x2013}$Poisson Equation
本文利用新的稳定性估计，将Hartree方程解的Wigner变换到Vlasov–Poisson方程解的$L^2$收敛速率从$\hbar^{3/4-\varepsilon}$改进为$\hbar$，并提出了类似多体Schrödinger方程平均场极限证明的新方法。
Global Existence and Time Decay for the Vlasov-Hartree System
研究弗拉索夫 - 哈特里系统的适定性和色散性质，用低正则泛函框架证明其全局适定性，粒子轨迹有意义，初始费米子密度正则时解是经典的，还给出相互作用为斥力和引力时系统的时间衰减及场增长情况。
Uniqueness criteria for the Vlasov--Poisson system and applications to semiclassical analysis
本文回顾了 Vlasov–Poisson 系统在强弱拓扑下的唯一性准则，并说明其如何指导半经典分析中不同量子态问题的求解。
Enhanced Stability in Quantum Optimal Transport Pseudometrics: From Hartree to Vlasov-Poisson
本文结合Golse和Paul的半经典最优传输距离方法及第一作者的新方法，在解具有有界密度的情况下，为Hartree动力学向Vlasov-Poisson方程的半经典极限建立了量子最优传输伪度量的几乎最优稳定性估计。
The semiclassical limit from Hartree to Vlasov at positive density: strong uniform-in-time convergence and scattering
研究正密度下从哈特里方程到弗拉索夫方程的半经典极限，通过证明有限时间区间收敛速率并结合相关界，得到全局存在和散射，量子与经典初始扰动对齐时收敛一致且有显式速率。
Solutions to a singular one-dimensional Vlasov equation are obtained as the semiclassical limit of the Wigner transform associated to a logarithmic Schrodinger equation. Two frameworks are considered, regarding in particular the initial position density: Gaussian initial density, or smooth initial density away from vacuum. For Gaussian initial densities, the analysis also yields global solutions to the isothermal Euler system that do not enter the frame of regular solutions to hyperbolic systems by P. D. Lax.
We consider mixed quasi-free states describing $N$ fermions in the mean-field limit. In this regime, the time evolution is governed by the nonlinear Hartree equation. In the large $N$ limit, we study the convergence towards the classical Vlasov equation. Under integrability and regularity assumptions on the initial state, we prove strong convergence in trace and Hilbert-Schmidt norm and provide explicit bounds on the convergence rate for a class of singular potentials of the form $V(x)=|x|^{-α}$, for $α\in(0,1/2)$.
A Gauge-Covariant Semiclassical Limit from the magnetic Hartree--Fock equation to the magnetic Vlasov--Poisson equation
研究三维空间中磁场存在下从磁Hartree--Fock方程到磁Vlasov--Poisson方程的半经典极限，建立规律性传播并利用规范协变磁Weyl量子化导出显式收敛界，结果具有规范协变性。
We justify WKB analysis for Hartree equation in space dimension at least three, in a regime which is supercritical as far as semiclassical analysis is concerned. The main technical remark is that the nonlinear Hartree term can be considered as a semilinear perturbation. This is in contrast with the case of the nonlinear Schrodinger equation with a local nonlinearity, where quasilinear analysis is needed to treat the nonlinearity.
From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials
本文研究费米子 N 体 Schrödinger 方程向奇异势 Hartree–Fock 与 Vlasov 方程的联合平均场和半经典极限，通过 h 一致正则性传播与 Schatten 范数估计给出推导及适用范围。
A common structure for modified scattering: Vlasov--Riesz, Hartree, and their coupling
本文发现Hartree、Vlasov--Riesz及其耦合方程共有的作用-密度结构，利用该结构在三维空间中建立了小初值全局存在性和修正散射，覆盖强长程势$\lambda|x|^{-\alpha}$（$0<\alpha\leq1/2$），并给出首个前向修正散射结果。
Propagation of moments for large data and semiclassical limit to the relativistic Vlasov equation
该研究推导半相对论Hartree-Fock方程的半经典极限，证明混合态下其以显式Schatten范数速率收敛于带奇异势的相对论Vlasov方程解的Weyl变换，推广了光滑势情形的已有结果，并给出该奇异相对论Vlasov方程适定性的新结论。
Global-in-time Semiclassical Regularity for the Hartree-Fock Equation
本文证明了在奇异相互作用下，Hartree-Fock 方程的解在任意长时间内保持关于半经典参数的一致正则性，并将从多体动力学推导 Hartree-Fock 和 Vlasov 方程的结论推广到任意长时间。
We consider the Wigner equation corresponding to a nonlinear Schroedinger evolution of the Hartree type in the semiclassical limit $\hbar\to 0$. Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in $L^2$ to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology. The proof makes use of a pointwise-positivity manipulation, which seems necessary in wor…
Long-time dynamics of Vlasov--Hartree systems across the Coulomb threshold
本文研究玻色-费米混合物的经典-量子模型——三维库仑Vlasov-Hartree系统，对小正则局域初始数据证明其整体适定性，明确其长时间动力学，结合动理学与色散方程方法，结果可推广至短程幂次逆相互作用。
We consider the semiclassical limit of nonlinear Schrödinger equations with wavepacket initial data. We recover the Wigner measure of the problem, a macroscopic phase-space density which controls the propagation of the physical observables such as mass, energy and momentum. Wigner measures have been used to create effective models for wave propagation in random media, quantum molecular dynamics, mean field limits, and the propagation of electrons in graphene. In nonlinear settings, the Vlasov-type equations obtained for the Wigner measure are often ill-posed on the physically interesting space…
We study the propagation of wave packets for nonlinear nonlocal Schrodinger equations in the semi-classical limit. When the kernel is smooth, we construct approximate solutions for the wave functions in subcritical, critical and supercritical cases (in terms of the size of the initial data). The validity of the approximation is proved up to Ehrenfest time. For homogeneous kernels, we establish similar results in subcritical and critical cases. Nonlinear superposition principle for two nonlinear wave packets is also considered.
A semiclassical limit of reduced Hartree-Fock theory at positive temperature
本文证明正温度下约化哈特里-福克巨势的半经典极限收敛于托马斯-费米巨势，提出其对偶表示并证明对偶问题的性质，所得结果与零温度情形及费米-狄拉克熵的标准泛函一致，还建立了一致的半经典迹公式。
Semiclassical Limit of the Bogoliubov-de Gennes Equation
本文建立了 Bogoliubov–de Gennes 方程的半经典极限，将其化为含有效平均场背景势的两粒子输运方程，并证明收敛性，同时结合已有结果给出自旋-1/2 费米子系统的联合半经典和平均场近似。
Semiclassical equivalence of two white dwarf models as ground states of the relativistic Hartree-Fock and Vlasov-Poisson energies
本文证明质量约束下相对论 Hartree–Fock 白矮星模型的基态能量和状态在半经典极限 $\hbar\to0$ 时收敛于相对论 Vlasov–Poisson 模型，从而揭示量子与经典平均场模型的等价性。
We consider a system of N trapped bosons with repulsive interactions in a combined semiclassical mean-field limit at positive temperature. We show that the free energy is well approximated by the minimum of the Hartree free energy functional - a natural extension of the Hartree energy functional to positive temperatures. The Hartree free energy functional converges in the same limit to a semiclassical free energy functional, and we show that the system displays Bose-Einstein condensation if and only if it occurs in the semiclassical free energy functional. This allows us to show that for weak …
We study nondegeneracy of ground states of the Hartree equation $$ -Δu+u=(I_{2}\ast u^2)u\quad\mbox{ in }\mathbb R^n $$ where $n=3,4,5$ and $I_2$ is the Newton potential. As an application of the nondegeneracy result, we use a Lyapunov-Schmidt reduction argument to construct multiple semiclassical solutions to the following Hartree equation with an external potential $$-\varepsilon^2Δu+u+V(x)u=\varepsilon^{-2}(I_{2}\ast u^2)u\quad \mbox{ in }\mathbb R^n.$$
The semi-classical regime of standing wave solutions of a Schrödinger equation in presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is show that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.
In this paper we study the semiclassical limit for the pseudo-relativistic Hartree equation $\sqrt{-\varepsilon^2 Δ+ m^2}u + V u = (I_α* |u|^{p}) |u|^{p-2}u$ in $\mathbb{R}^N$ where $m>0$, $2 \leq p < \frac{2N}{N-1}$, $V \colon \mathbb{R}^N \to \mathbb{R}$ is an external scalar potential, $I_α(x) = \frac{c_{N,α}}{|x|^{N-α}}$ is a convolution kernel, $c_{N,α}$ is a positive constant and $(N-1)p-N<α<N$. For $N=3$, $α=p=2$, our equation becomes the pseudo-relativistic Hartree equation with Coulomb kernel.
Modified scattering for semiclassical Bose-Fermi mixtures
本文针对Vlasov-Hartree系统，证明了小初始数据下玻色-费米混合体系存在修正散射，并利用混合拉格朗日与傅里叶方法给出了渐近动力学的显式描述，且无需对初始费米子密度求导。
We consider the Hartree equation with a smooth kernel and an external potential, in the semiclassical regime. We analyze the propagation of two initial wave packets, and show different possible effects of the interaction, according to the size of the nonlinearity in terms of the semiclassical parameter. We show three different sorts of nonlinear phenomena. In each case, the structure of the wave as a sum of two coherent states is preserved. However, the envelope and the center (in phase space) of these two wave packets are affected by nonlinear interferences, which are described precisely.
For the gravitational Vlasov-Poisson equation, Guo and Rein constructed a class of classical isotropic states as minimizers of free energies (or energy-Casimir functionals) under mass constraints. For the quantum counterpart, that is, the gravitational Hartree equation, isotropic states are constructed as free energy minimizers by Aki, Dolbeault and Sparber. In this paper, we are concerned with the correspondence between quantum and classical isotropic states. Precisely, we prove that as the Planck constant $\hbar$ goes to zero, free energy minimizers for the Hartree equation converge to those…
The Pauli-Poisson equation and its semiclassical limit
本研究针对泡利-泊松方程，证明其在能量空间的全局适定性，借助维格纳变换与密度矩阵方法推导其到带洛伦兹力的磁弗拉索夫-泊松方程的半经典极限，拓展了非相对论情形的相关研究。
We consider the three-dimensional semi-relativistic Hartree model for fast quantum mechanical particles moving in a self-consistent field. Under appropriate assumptions on the initial density matrix as a (fully) mixed quantum state we prove, using Wigner transformation techniques, that its classical limit yields the well known relativistic Vlasov-Poisson system. The result holds for the case of attractive and repulsive mean-field interaction, with an additional size constraint in the attractive case.
Semi-classical limit of quantum scattering states for the nonlinear Hartree equation
研究半经典区域非线性Hartree方程量子散射态的长时间动力学，通过新的自由薛定谔流一致色散估计，证明其Wigner变换半经典极限弱*收敛于Vlasov方程经典散射态，顺带建立无初值正则性假设的Vlasov方程小初值散射结果。
In this paper the Hartree equation is derived from the $N$-body Schr\''odinger equation in the mean-field limit, with convergence rate estimates that are uniform in the Planck constant $\hbar$. Specifically, we consider the two following cases:(a) T\''oplitz initial data and Lipschitz interaction forces, and (b) analytic initial data and interaction potential, over short time intervals independent of $\hbar$. The convergence rates in these two cases are $1/\sqrt{\log\log N}$ and $1/N$ respectively. The treatment of the second case is entirely self-contained and all the constants appearing in t…
We consider the semiclassical limit of the Hartree equation with a data causing a focusing at a point. We study the asymptotic behavior of phase function associated with the WKB approximation near the caustic when a nonlinearity is supercritical. In this case, it is known that a phase shift occurs in a neighborhood of focusing time in the case of focusing cubic nonlinear Schrödinger equation. Thanks to the smoothness of the nonlocal nonlinearities, we justify the WKB-type approximation of the solution for a data which is larger than in the previous results and is not necessarily well-prepared.…
In this paper, we establish (1) the classical limit of the Hartree equation leading to the Vlasov equation, (2) the classical limit of the $N$-body linear Schrödinger equation uniformly in N leading to the N-body Liouville equation of classical mechanics and (3) the simultaneous mean-field and classical limit of the N-body linear Schrödinger equation leading to the Vlasov equation. In all these limits, we assume that the gradient of the interaction potential is Lipschitz continuous. All our results are formulated as estimates involving a quantum analogue of the Monge-Kantorovich distance of ex…
Phase mixing for the Hartree equation and Landau damping in the semiclassical limit
研究Hartree方程在平移不变稳态附近的渐近行为，通过相位混合估计证明密度衰减和散射，提供朗道阻尼的量子类比，并在半经典极限下一致成立。
Derivation of the Vlasov-Maxwell system from the Maxwell-Schrödinger equations with extended charges
该研究针对库仑规范下带扩展电荷的Maxwell-Schrödinger方程，在半经典区域证明其收敛于非相对论Vlasov-Maxwell动力学并给出近似精度估计，同时建立了两类方程的适定性与正则性理论。
Derivation of the Maxwell-Schrödinger and Vlasov-Maxwell Equations from Non-Relativistic QED
该论文从非相对论量子电动力学出发，在平均场极限下推导出麦克斯韦-薛定谔方程和弗拉索夫-麦克斯韦方程，证明了多体量子态收敛于半经典Slater行列式与相干光子态的乘积。
Semiclassical limit of cubic nonlinear Schrödinger equations for mixed states
本文证明混合态三次非线性薛定谔方程在半经典极限下收敛到奇异Vlasov方程，并给出量子Penrose稳定性条件，该条件对小数据及散焦情形下的物理相关扰动均成立。
Nonlinear Schrodinger Equations (NLS) of the Hartree type occur in the modeling of quantum semiconductor devices. Their "semiclassical" limit of vanishing (scaled) Planck constant is both a mathematical challenge and practically relevant when coupling quantum models to classical models. With the aim of describing the semi-classical limit of the 3D Schrodinger--Poisson system with an additional harmonic potential, we study some semi-classical limits of the Hartree equation with harmonic potential in space dimension n>1. The harmonic potential is confining, and causes focusing periodically in ti…
In two-dimensional space there are possibilities for quantum statistics continuously interpolating between the bosonic and the fermionic one. Quasi-particles obeying such statistics can be described as ordinary bosons and fermions with magnetic interactions. We study a limit situation where the statistics/magnetic interaction is seen as a "perturbation from the fermionic end". We vindicate a mean-field approximation, proving that the ground state of a gas of anyons is described to leading order by a semi-classical, Vlasov-like, energy functional. The ground state of the latter displays anyonic…
Global derivation of the 1D Vlasov-Poisson equation from quantum many-body dynamics with screened Coulomb potential
研究带屏蔽库仑势的一维量子多体动力学，通过结合量子平均场、半经典和德拜长度极限，证明了一维 Vlasov-Poisson 方程的全局推导，并解决了相关守恒律开放问题的一维情形。
The semiclassical limit from the Pauli-Poisswell/Darwin to the Euler-Poisswell/Darwin system by WKB methods
本文通过 WKB 方法研究了从泡利-泊松威尔/达尔文到欧拉-泊松威尔/达尔文系统的 semiclassical 极限，证明了时间局部的 semiclassical 极限和弱收敛性，以及宏观密度的强收敛性。
Normalized Solutions and Semiclassical Concentration for Upper-Critical Fractional Choquard Equations
本文研究了带有 $L^2$-超临界扰动的上临界分数Choquard方程归一化解的多重性，通过约束变分方法证明了在小参数下存在多个集中解。
Multiphasic formulation of Vlasov equations and applications
本文研究Vlasov方程的多相公式，将其化为耦合无压力Euler系统，在低正则性下建立局部适定性统一理论，并应用于Vlasov-Poisson与Vlasov-Navier-Stokes系统，证明极限与稳定性结果。
On the effective dynamics of Bose-Fermi mixtures
本文研究零温下玻色-费米混合体系的动力学，先证明平均场近似下耦合Hartree型方程的涌现，再提出费米子半经典、玻色子量子化的新标度机制，并证明向耦合Vlasov-Hartree系统的显式收敛速率。
We consider the N-body Schrödinger dynamics of bosons in the mean field limit with a bounded pair-interaction potential. According to the previous work \cite{AmNi}, the mean field limit is translated into a semiclassical problem with a small parameter $ε\to 0$, after introducing an $ε$-dependent bosonic quantization. The limit is expressed as a push-forward by a nonlinear flow (e.g. Hartree) of the associated Wigner measures. These object and their basic properties were introduced in \cite{AmNi} in the infinite dimensional setting. The additional result presented here states that the transport…
The quantitative semi-classical limit of a large Fermi system at zero temperature
本文研究三维大型费米子系统在平均场与半经典极限下基态 Wigner 函数向经典 Thomas-Fermi 理论的收敛，主要贡献是量化了关于半经典参数的收敛速率，并涵盖了直至 Coulomb 相互作用的奇异势。
This course explains how the usual mean field evolution partial differential equations (PDEs) in Statistical Physics - such as the Vlasov-Poisson system, the vorticity formulation of the two-dimensional Euler equation for incompressible fluids, or the time-dependent Hartree equation in quantum mechanics - can be rigorously derived from first principles, i.e. from the fundamental microscopic equations that govern the evolution of large, interacting particle systems. The emphasis is put on the mathematical methods used in these derivations, such as Dobrushin's stability estimate in the Monge-Kan…
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