Derivation of the Gross-Pitaevskii Hierarchy
Benjamin Schlein
本文报道了在大量粒子的限制下，被囚禁的玻色-爱因斯坦凝聚体的动力学行为。这些结果是在与L. Erd \H os和H.- T.你。
We report on some recent results regarding the dynamical behavior of a trapped Bose-Einstein condensate, in the limit of a large number of particles. These results were obtained in \cite{ESY}, a joint work with L. Erd\H os and H.-T. Yau.
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The Gross-Pitaevskii functional with a random background potential and condensation in the single particle ground state
对于具有随机背景势的离散和连续Gross-Pitaevskii能量泛函，我们研究了Gross-Pitaevskii基态.当Gross-Pitaevskii基态和随机背景哈密顿量的基态渐近重合时，我们描述了一种相互作用耦合机制.
The paper is concerned with the existence and blow-up behavior of the minimizers for the 2D attractive Gross-Pitaevskii functional when the interaction strength increases to a critical value. Our results hold for all bounded external potential satisfying some general assumptions.
On the Gross-Pitaevskii equation for trapped dipolar quantum gases
研究了描述囚禁偶极量子气体玻色-爱因斯坦凝聚的含时Gross-Pitaevskii方程。研究了解的存在唯一性和可能的爆破性。此外，我们还讨论了这种非线性非局部薛定谔方程的降维问题。
Semiclassical symmetry of the Gross-Pitaevskii equation with quadratic nonlocal Hamiltonian
本文将具有二次非局部非线性项的Gros-Pitaevsky方程的Cauchy问题化为相应的线性方程的Cauchy问题。本文讨论了线性和非线性Gros-Pitaevsky方程的对称算子之间的关系。
The effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii equation within the "projection counting" scheme.
Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation
我们考虑的积分微分多维Gross-Pitaevskii方程的对称性与非局部非线性（立方）项的对称性分析的背景下使用的形式主义的半经典渐近。这产生了一个半经典约化的非局部Gross-Pitaevskii方程，它可以被视为一个近似线性方程，以确定半经典渐近解的主项。我们的主要结果是一种方法，它允许一个构造一类对称算子的减少Gross-Pitaevskii方程。这些对称性算子是由包括交织算子和附加代数条件在内的线性关系确定的。用一维简化Gross-Pitaevskii方程说明了基本思想。明确地找到了对称算子，并得到了相应的精确解族。
We consider systems of bosons trapped in a box, in the Gross-Pitaevskii regime. We show that low-energy states exhibit complete Bose-Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in \cite{BBCS1}, removing the assumption of small interaction potential.
Rigorous Derivation of the Gross-Pitaevskii Equation
含时Gross-Pitaevskii方程描述了初始捕获的玻色-爱因斯坦凝聚体的动力学。我们提出了一个严格的证明，这一事实从一个多体玻色子薛定谔方程与短尺度排斥相互作用的稀释极限。我们的证明表明，一个明确的短尺度相关结构的凝聚体的持久性。
Derivation of the 3D quintic Gross--Pitaevskii equation
本文推导了描述三维三体相互作用玻色-爱因斯坦凝聚体的五次格罗斯-皮塔耶夫斯基方程，并证明了该方程的有效耦合常数与三体散射超体积有关。
Berry phases for the nonlocal Gross-Pitaevskii equation with a quadratic potential
利用复芽方法在绝热近似下构造了具有二次位势和非局部非线性项的非定常Gros-Pitaevskii方程的可数个渐近空间定域解集.渐近参数是1/T，其中$T\gg1$是绝热演化时间。将线性薛定谔方程的Berry相位推广到Gross-Pitaevskii方程。对于构造的解决方案，Berry相位被发现在明确的形式。
We consider a gas of interacting bosons trapped in a box of side length one in the Gross-Pitaevskii limit. We review the proof of the validity of Bogoliubov's prediction for the ground state energy and the low-energy excitation spectrum. This note is based on joint work with C. Brennecke, S. Cenatiempo and B. Schlein.
Diagonalizing Bose Gases in the Gross-Pitaevskii Regime and Beyond
本文提出一种新的 Bogoliubov 理论方法，在 Gross-Pitaevskii 区域初等推导 Lee-Huang-Yang 公式，并推广低激发谱至更奇异标度，给出基态能量上界。
On the Cauchy problem for Gross-Pitaevskii hierarchies
本文研究了$\mathbb{R}^n,$上Gross-Pitaevskii无穷线性方程族的Cauchy问题，证明了具有$α> n/2.$的边际密度算子序列在Sobolev型空间$\mathrm{H}^α_ξ$中解的局部存在唯一性，特别地，我们给出了$α> n/2,$的所有情形的清楚的讨论，它涵盖了在这种情形下Gross-Pitaevskii方程族的局部适定性问题.
We present a microscopic derivation of the time-dependent Gross-Pitaevskii equation starting from an interacting N-particle system of Bosons. We prove convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the respective Gross-Pitaevskii equation. Our work extends a previous result by one of us (P.P.[44]) to interaction potentials which need not to be nonnegative, but may have a sufficiently small negative part. One key estimate in our proof is an operator inequality which was first proven by Jun Yin, see [49].
We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. Under a physically motivated assumption on the energy of the initial data, we show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross-Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e. on the number of excitations of the Bose-Einstein condensate). To reach this goal, we combine the method of \cite{LNS}, where fluctuations a…
Derivation of the Time Dependent Gross Pitaevskii Equation with External Fields
使用一种新的方法[9]，可以从相互作用粒子的微观N体薛定谔演化中导出平均场方程，而不使用BBGKY系列。最近，这种方法被用来推导奇异相互作用的Hartree方程[4]和相互作用上没有正性条件的Gross Pitaevskii方程[10]，其中必须限制相互作用的标度行为。在本文中，更一般的标度应考虑假设积极的相互作用。
Deriving the Gross-Pitaevskii equation
在实验中，玻色-爱因斯坦凝聚体是通过在阱中冷却稀玻色气体来制备的。在达到相变之后，关闭阱并观察冷凝物的演变。宏观上用Gross-Pitaevskii方程描述了这种演化。在微观水平上，玻色气体的动力学由$N$体薛定谔方程描述。我们在文献[BdS 12]中构造了一类在Fock空间中能量上接近于基态的初始数据，并证明了它们的演化近似遵循Gross-Pitaevskii方程.其核心思想是用Bogoliubov变换来模拟两粒子关联。
Local well-posedness for Gross-Pitaevskii hierarchies
考虑$\mathbb{R}^n.$上Gross-Pitaevskii无穷线性方程族的Cauchy问题.通过在边缘密度矩阵序列的Sobolev型空间中引入（F）-范数，得到了解的局部存在唯一性和稳定性.我们也得到了解的显式时空型估计。特别地，这个（F）-范数与通常的Sobolev空间范数相容，只要初始数据被分解。
Expansion of the energy of the ground state of the Gross-Pitaevskii equation in the Thomas-Fermi limit
由Gallo和Pelinovsky在Thomas-Fermi极限下给出的Gross-Pitaevskii方程的基态的渐近展开式，我们导出了基态的动能、势能和总能量的渐近展开式.特别地，我们给出了Dalfovo，Pitaevskii和Stringari在空间维数为3的情况下计算的动能展开式的严格证明。此外，我们在这个展开式中计算了一个额外的项，并将结果推广到空间维度1和2。
Quasi-periodic solutions of the Gross-Pitaevskii equation with a periodic potential in dimension three are studied. It is proven that there is an extensive "non-resonant" set ${\mathcal G} \subset \mathbb{R}^3$ such that for every $\vec k\in \mathcal G$ there is a solution asymptotically close to a plane wave $Ae^{i\langle{ \vec{k}, \vec{x} }\rangle}$ as $|\vec k|\to \infty $, given $A$ is sufficiently small.
In this paper, we revise the derivation of the Gross-Pitaevskii equation from the Heisenberg spin chain stressing physical aspects and all technicalities.
Exponential Control of Excitations for Trapped BEC in the Gross-Pitaevskii Regime
本文研究了三维陷俘 Bose 气体在 Gross-Pitaevskii 极限下的 Bose-Einstein 凝聚现象，证明了正交于凝聚态的粒子数量的指数控制，推广了平移不变系统的已有结果。
We show that Bogoliubov theory correctly predicts the low-energy spectral properties of Bose gases in the Gross-Pitaevskii regime. We recover recent results from \cite{BBCS,BBCS4}. While our main strategy is similar to the one developed in \cite{BBCS,BBCS4}, we combine it with new ideas, taken in part from \cite{Hainzl,NT}; this makes our proof substantially simpler and shorter. As an important step towards the proof of Bogoliubov theory, we show that low-energy states exhibit complete Bose-Einstein condensation with optimal control on the number of orthogonal excitations.
Quasi-periodic solutions of a nonlinear polyharmonic equation for the case $4l>n+1$ in $\R^n$, $n>1$, are studied. This includes Gross-Pitaevskii equation in dimension two ($l=1,n=2$). It is proven that there is an extensive "non-resonant" set ${\mathcal G}\subset \R^n$ such that for every $\vec k\in \mathcal G$ there is a solution asymptotically close to a plane wave $Ae^{i\langle{ \vec{k}, \vec{x} }\rangle}$ as $|\vec k|\to \infty $, given $A$ is sufficiently small.
On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
我们给出了一个新的证明的唯一性的解决方案的Gross-Pitaevskii的层次，首先建立了由Erdos，Schlein和丘，在不同的空间，基于时空估计。
We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson's upper bound, which provides the correct leading order asymptotics for hard spheres. Afterwards, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross-Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross-Pitaeavskii limit.…
We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order $N^{-1+κ}$, for $κ>0$. Assuming that $κ\in (0;1/43)$, we show that low-energy states of the system exhibit complete Bose-Einstein condensation by providing explicit bounds on the expectation and on higher moments of the number of excitations.
Ermakov-Lewis Invariants of the Gross-Pitaevskii Equation
本文研究非线性Gross-Pitaeviskii方程的Ermakov-Lewis不变量
Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions
我们研究了两个和三个空间维度的Gross-Pitaevskii方程在时间无穷远处接近非零常数平衡点的解的渐近行为。我们构造了一类具有指定色散渐近行为的整体解，该解是根据线性化发展给出的。
Interacting Brownian motions and the Gross-Pitaevskii formula
我们回顾概率方法描述相互作用的粒子稀系统的Gross-Pitaevskii理论。主要成果是在囚禁势中相互作用布朗运动大系统的平均占有测度的大偏差原理。相应的速率函数作为变分问题给出，其解提供了无限系统的有效描述。
Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases
研究了描述偶极量子气体玻色-爱因斯坦凝聚的Gross-Pitaevskii方程驻波解的稳定性，并刻画了它们的轨道。作为中间步骤，我们考虑相应的约束极小化问题，并建立基态解的存在性，对称性和唯一性。
Bäcklund transformations are applied to study the Gross-Pitaevskii equation. Supported by previous results, a class of Bäcklund transformations admitted by this equation are constructed. Schwartzian derivative as well as its invariance properties turn out to represent a key tool in the present investigation. Examples and explicit solutions of the Gross-Pitaevskii equation are obtained.
Solutions of the Gross-Pitaevskii and time-fractional Gross-Pitaevskii equations for different potentials with Homotopy Perturbation Method
本文在简要介绍了标准Gross-Pitaevskii方程的基础上，提出了分数阶Gross-Pitaevskii方程来研究弱相互作用玻色粒子系统的玻色-爱因斯坦凝聚的含时基态动力学。我们只关注时间分数阶Gross-Pitaevskii方程，利用同伦微扰方法，在外陷阱势$V(x)=0$和光晶格势$V(x) =\pm\sin^{2}x$的情况下，得到了标准Gross-Pitaevskii方程和时间分数阶Gross-Pitaevskii方程的吸引力和排斥力的解。我们发现，对于这些势和相互作用，Gross-Pitaevskii方程的同伦摄动方法解与其解析解是相同的，而且我们还发现，对于这些势和相互作用，时间分数阶Gross-Pitaevskii方程的解可以用Mittag-Leffler函数给出.时间分数阶Gross-Pitaevskii方程的解表明，玻色粒子的玻色-爱因斯坦凝聚的基态动力学随时间的演化偏离指数形式，并以指数形式随时间演化.
We consider systems of $N$ bosons in $\mathbb{R}^3$, trapped by an external potential. The interaction is repulsive and has a scattering length of the order $N^{-1}$ (Gross-Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum up to errors that vanish in the limit $N\to \infty$.
Scattering for the Gross-Pitaevskii equation
本文研究了Gross-Pitaevskii（或Ginzburg-Landau Schroedinger）方程在时间无穷远时非零常数平衡点附近解的渐近行为。证明了在大于3维的空间中，小扰动在无穷远时刻可以用线性化演化来逼近，并且在某些索伯列夫空间中，波算子在0附近是同胚的。
We review some mathematical work on the Bose gas in the Gross-Pitaevskii regime. We start with the classical results by Lieb, Seiringer and Yngvason on the ground state energy and by Lieb and Seiringer on the existence of Bose-Einstein condensation. Afterwards, we discuss some more recent progress, based on a rigorous version of Bogoliubov theory.
We prove an adiabatic theorem for the non-autonomous Gross-Pitaevskii equation in the case of a weak trap. More precisely, we assume that the external potential decays suitably at infinity and admits exactly one bound state.
We consider Bose gases consisting of $N$ particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of the order $N^{-1}$(Gross-Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as $N \to \infty$. Our results confirm Bogoliubov's predictions.
Vortex dynamics for the two dimensional non homogeneous Gross-Pitaevskii equation
在薛定谔动力学下，当Ginzburg-Landau参数趋于零时，我们导出了非均匀背景密度中Ginzburg-Landau涡的渐近动力学定律。新的成分涉及整个核心的下界估计和近似。
Madelung, Gross-Pitaevskii and Korteweg
本文综述了非线性薛定谔方程的流体动力学表述的各个方面，通过Madelung变换，在connections的量子流体动力学模型和可压缩流体的Korteweg型。
Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional
在最近的实验中，考虑到在外部势中相互作用玻色气体的基态性质，通常通过Gross-Pitaevskii能量泛函来描述。本文首次证明了这种近似对具有正相互作用的稀玻色气体的基态能量和粒子密度的渐近精确性。
We consider Bose gases of $N$ particles in a box of volume one, interacting through a repulsive potential with scattering length of order $N^{1-κ}$, for $κ> 0$. Such regimes interpolate between the Gross-Pitaevskii and thermodynamic limits. Assuming that $κ$ is sufficiently small, we determine the ground state energy and the low-energy excitation spectrum of the system, up to errors vanishing in the limit of large $N$.
Bounds on the tight-binding approximation for the Gross-Pitaevskii equation with a periodic potential
我们证明的有效性的离散非线性薛定谔方程的紧束缚近似的上下文中的Gross-Pitaevskii方程与周期性的潜力。我们的建设周期性的潜力和相关的Wannier函数是基于以前的工作，而我们的分析涉及能源估计和Gronwall的不等式地址的时间依赖性本地化的解决方案，但大的有限的时间间隔。
Global well-posedness of the Gross--Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions
考虑了空间无穷远处具有非零边界条件的$\R^4$上的Gros-Pitaevskii方程和空间无穷远处具有非零边界条件的$\R^3$上的五阶非线性薛定谔方程.通过将这些方程看作对能量临界NLS的扰动，我们证明了它们在能量空间中是全局适定的。特别地，我们证明了这些方程在能量空间中的无条件唯一性。
Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti
本文给出了$\R^3$中三次Gross-Pitaevskii族解的无条件唯一性的一个新的、更简单的证明。我们分析的主要工具之一是量子de Finetti定理。我们的唯一性结果与Erdös，Schlein和Yau的著名著作\cite{esy1,esy2,esy3,esy4}中的结果是等价的.
Blowup alternative for Gross-Pitaevskii hierarchies
本文证明了$\mathbb{R}^n$上Gross-Pitaevskii族的blow-up择一性，并给出了相应的blow-up率的下界.特别地，我们证明了当初始条件下某些$k$粒子的能量为负时，对于$n \ge 3$，聚焦Gross-Pitaevskii族的密度算符的任何解在有限时间内爆破.所有这些结果都成立，而无需对初值和容许初值作因子分解条件的假设。我们的分析是基于使用的准Banach空间的序列的边缘密度矩阵。
Solutions of the classical and nonlocal Gross-Pitaevskii (GP) equation with a parabolic potential and a gain term are derived by using a second order nonisospectral Ablowitz-Kaup-Newell-Segur system and reduction technique of double Wronskians. Solutions of the classical GP equation show typical space-time localized characteristics. An interesting dynamics, solitons carrying an oscillating wave, are found with mathematical analysis and illustrations. Solutions of some nonlocal cases are also illustrated.
On the well-posedness and scattering for the Gross-Pitaevskii hierarchy via quantum de Finetti
证明了在${\mathbb R}^3$中散焦立方Gross-Pitaevskii（GP）族的散射态的存在性.此外，我们表明，通常用于GP层次的适定性理论的能量增长条件是，在特定意义上，必要的。事实上，我们证明，没有后者，存在初始数据的聚焦立方GP的层次，瞬时爆破发生。
In this paper, we study solutions to the three-dimensional quintic Gross-Pitaevskii hierarchy. We prove unconditional uniqueness among all small solutions in the critical space $\mathfrak{H}^1$ (which corresponds to $H^1$ on the NLS level). With slight modifications to the proof, we also prove unconditional uniqueness of solutions to the Hartree hierarchy without smallness condition. Our proof uses the quantum de Finetti theorem, and is an extension of the work by Chen-Hainzl-Pavlović-Seiringer \cite{CHPS}, and our previous work \cite{UniqueLowReg}.
We consider the (de)focusing cubic Gross-Pitaevskii (GP) hierarchy on $\mathbb{R}$, which is an infinite hierarchy of coupled linear inhomogeneous PDE which appears in the derivation of the cubic nonlinear Schrödinger (NLS) equation from quantum many-particle systems. Motivated by the fact that the cubic NLS on $\mathbb{R}$ is an integrable equation which admits infinitely many conserved quantities, we exhibit an infinite sequence of operators which generate analogous conserved quantities for the GP hierarchy.
We consider a gas of $N$ bosons in a box with volume one interacting through a two-body potential with scattering length of order $N^{-1}$ (Gross-Pitaevskii limit). Assuming the (unscaled) potential to be sufficiently small, we show that the ground state of the system and all states with relatively small excitation energy exhibit complete Bose-Einstein condensation, with a uniform (i.e. $N$ independent) bound on the number of excitations.
Unconditional Uniqueness of the cubic Gross-Pitaevskii Hierarchy with Low Regularity
本文在低正则性Sobolev型空间中建立了$\mathbb{R}^d$上三次Gross-Pitaevskii族解的无条件唯一性.更准确地说，我们减少了正则性$s$到目前已知的三次非线性薛定谔方程的解的无条件唯一性的正则性要求（$s\ge\frac{d}{6}$，如果$d=1,2$和$s>s_c=\frac{d-2}{2}$，如果$d\ge 3$）。这样，我们推广了Chen-Hainzl-Pavlović-Seiringer最近的工作。
We study a dilute gas of interacting fermions at temperature $T=0$ and chemical potential $μ\in \mathbb{R}$. The particles are trapped by an external potential, and they interact via a microscopic attractive two-body potential with a two-body bound state. We prove the emergence of the macroscopic Gross-Pitaevskii theory as first-order contribution to the BCS energy functional in the regime of vanishing micro-to-macro scale parameter.
Standing waves of the Anderson-Gross-Pitaevskii equation
本文研究了安德森-格罗斯-皮塔耶夫斯基方程在1维和2维中的驻波构造及其正则性、局部化和稳定性特性。
We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose-Einstein condensation, with almost optimal bounds on the number of orthogonal excitations.
We study the Bose-Einstein condensates of trapped Bose gases in the Gross-Pitaevskii regime. We show that the ground state energy and ground states of the many-body quantum system are correctly described by the Gross-Pitaevskii equation in the large particle number limit, and provide the optimal convergence rate. Our work extends the previous results of Lieb, Seiringer and Yngvason on the leading order convergence, and of Boccato, Brennecke, Cenatiempo and Schlein on the homogeneous gas. Our method relies on the idea of 'completing the square', inspired by recent works of Brietzke, Fournais an…
This article consists of two parts. In the first part, we review the most recent proofs establishing quadratic Morawetz inequalities for the nonlinear Schrödinger equation (NLS). We also describe the applications of these estimates to the problem of quantum scattering. In the second part, we generalize some of the methods developed for the NLS by many authors to the case of Gross-Pitaevskii (GP) hierarchies. In particular, we prove both regular and interaction Morawetz identities for the GP hierarchy, which appear here for the first time in the literature.
We perform a rigorous analysis of the anelastic approximation for the Gross-Pitaevskii equation with $x$-dependent chemical potential. For general initial data and periodic boundary condition, we show that as $\eps\to 0$, equivalently the Planck constant tends to zero, the density $|ψ^{\eps}|^{2}$ converges toward the chemical potential $ρ_{0}(x)$ and the velocity field converges to the anelastic system. When the chemical potential is a constant, the anelastic system will reduce to the incompressible Euler equations. The resonant effects the singular limit process and it can be overcome becaus…
Energy conservation and blowup of solutions for focusing Gross-Pitaevskii hierarchies
我们认为解决方案的重点三次和五次Gross-Pitaevskii（GP）的层次。我们确定了一个可观察到的对应于每个粒子的平均能量，我们证明了它是一个守恒量。证明了当初始条件下每个粒子的能量为负时，在$L^2$ -临界或$L^2$ -超临界水平上聚焦GP族的所有解在有限时间内爆破.我们的结果不假设任何分解的初始数据。
A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies
我们考虑三次和五次Gross-Pitaevskii（GP）层次结构中的$d\geq1$维度，聚焦和散焦的相互作用。我们提出了一个新的证明的存在性的解决方案，不需要先验约束的时空规范，这是介绍了在工作中的Klainerman和Machedon，\cite{klma}，并在我们早期的工作，\cite{chpa2}。
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