我们提出了一个微观推导的含时Gross-Pitaevskii方程从相互作用的N粒子系统的玻色子。我们证明了收敛的减少密度矩阵对应的精确时间演化到投影到各自的Gross-Pitaevskii方程的解决方案。我们的工作推广了我们之一（P. P. [44])相互作用势不一定非负，但可以有足够小的负部分。我们证明中的一个关键估计是一个算子不等式，它首先由Jun Yin证明，见[49]。
英文摘要：
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].
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.
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 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.
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 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 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 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.
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…
