多粒子Anderson局域化：粒子数的归纳
Multi-Particle Anderson Localisation: Induction on the Number of Particles
摘要：
本文是我们最近关于两粒子Anderson模型的论文\cite{CS08}和\cite{CS09}的后续。在这里，我们建立的量子$N$粒子系统的安德森局域化现象的晶格$\Z^d$与短程相互作用，并在存在的IID外部潜力与足够经常的边际累积分布函数（CDF）。我们的主要方法是多尺度分析（MSA; cf. \cite{FS}，\cite{FMSS}，\cite{DK}）到多粒子系统，结合粒子数的归纳，如我们早期的手稿\cite{CS07}中所提出的。最近，Aizenman和Warzel \cite{AW08}在一项独立的工作中也得到了类似的结果：他们提出了一种扩展的分数矩方法（FMM），该方法是在\cite{AM93}和\cite{ASFH01}（也见其中的参考文献）中为单粒子模型开发的，也结合了粒子数的归纳。在我们的证明中的一个重要作用是由Stollmann的本征值浓度界的一个变体（参见。\cite{St00}）。正如前面在\cite{C08}中所证明的那样，这个结果可以直接扩展到具有相关外部随机势的多粒子系统的情况：这是我们未来工作的一个主题。我们还强调，我们的证明方案是\textit{not}特定的晶格系统，因为我们的主要方法，MSA，承认一个连续的版本。在具有各种类型的外部随机势的连续相互作用系统中，多粒子Anderson局域化的证明将在单独的论文中发表。
英文摘要：
This paper is a follow-up of our recent papers \cite{CS08} and \cite{CS09} covering the two-particle Anderson model. Here we establish the phenomenon of Anderson localisation for a quantum $N$-particle system on a lattice $\Z^d$ with short-range interaction and in presence of an IID external potential with sufficiently regular marginal cumulative distribution function (CDF). Our main method is an adaptation of the multi-scale analysis (MSA; cf. \cite{FS}, \cite{FMSS}, \cite{DK}) to multi-particle systems, in combination with an induction on the number of particles, as was proposed in our earlier manuscript \cite{CS07}. Similar results have been recently obtained in an independent work by Aizenman and Warzel \cite{AW08}: they proposed an extension of the Fractional-Moment Method (FMM) developed earlier for single-particle models in \cite{AM93} and \cite{ASFH01} (see also references therein) which is also combined with an induction on the number of particles. An important role in our proof is played by a variant of Stollmann's eigenvalue concentration bound (cf. \cite{St00}). This result, as was proved earlier in \cite{C08}, admits a straightforward extension covering the case of multi-particle systems with correlated external random potentials: a subject of our future work. We also stress that the scheme of our proof is \textit{not} specific to lattice systems, since our main method, the MSA, admits a continuous version. A proof of multi-particle Anderson localization in continuous interacting systems with various types of external random potentials will be published in a separate papers.
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