考虑一个三维$N$玻色子系统通过一个排斥短程对势$N^2V(N(x_i-x_j))$相互作用，其中$\bx=(x_1, >..., x_N)$表示粒子的位置。令$H_N$表示系统的哈密顿量，令$ψ_{N,t}$表示薛定谔方程的解。假设初始数据$ψ_{N,0}$满足能量条件\[ < ψ_{N,0}, H_N ψ_{N,0} > \leq C N >. \]，并且单粒子密度矩阵收敛到投影为$N \to \infty$。证明了$ψ_{N,t}$的$k$ -粒子密度矩阵在极限$N \to \infty$中分解.此外，单粒子轨道波函数求解了含时的Gross-Pitaevskii方程，这是一个立方非线性薛定谔方程，耦合常数与势能$V$的散射长度成正比。在\cite{ESY}，我们证明了相同的声明的条件下，相互作用势$V$是足够小的，在目前的工作中，我们开发了一种新的方法，不需要限制的潜力的大小。
英文摘要：
Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $ψ_{N,t}$ be the solution to the Schrödinger equation. Suppose that the initial data $ψ_{N,0}$ satisfies the energy condition \[ < ψ_{N,0}, H_N ψ_{N,0} > \leq C N >. \] and that the one-particle density matrix converges to a projection as $N \to \infty$. Then, we prove that the $k$-particle density matrices of $ψ_{N,t}$ factorize in the limit $N \to \infty$. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant proportional to the scattering length of the potential $V$. In \cite{ESY}, we proved the same statement under the condition that the interaction potential $V$ is sufficiently small; in the present work we develop a new approach that requires no restriction on the size of the potential.
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.
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 $N\to \infty$.
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 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.
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 $N$.
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.
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…
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.
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.
In this paper, we revise the derivation of the Gross-Pitaevskii equation from the Heisenberg spin chain stressing physical aspects and all technicalities.
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.
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.
We study the dipolar Gross-Piteavskii functional with the Lee-Huang-Yang (LHY) correction term without trapping potential and in the regime where the dipole-dipole interaction dominates the repulsive short-range interaction. We show that, above a critical mass, the functional admits minimizers and we prove their regularity and exponential decay. We also estimate the critical mass in terms of the parameters of the system.
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…
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.
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.
We study a generalized dissipative Gross-Pitaevskii-type model arising in the description of exciton-polariton condensates. We derive global in-time existence results and various a-priori estimates for this model posed on the one-dimensional torus. Moreover, we analyze in detail the long-time behavior of spatially homogenous solutions and their respective steady states and present numerical simulations in the case of more general initial data. We also study the convergence to the corresponding adiabatic regime, which results in a single damped-driven Gross-Pitaveskii equation.
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.
