Multi-particle Anderson Localisation: Induction on the Number of Particles

Abstract

This paper is a follow-up of our recent papers Chulaevsky and Suhov (Commun Math Phys 283:479–489, 2008) and Chulaevsky and Suhov (Commun Math Phys in press, 2009) covering the two-particle Anderson model. Here we establish the phenomenon of Anderson localisation for a quantum N-particle system on a lattice \(\mathbb Z^d\) 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. Fröhlich and Spencer, Commun Math Phys 88:151–184, 1983; Fröhlich et al., Commun Math Phys 101:21–46, 1985; von Dreifus and Klein, Commun Math Phys 124:285–299, 1989) to multi-particle systems, in combination with an induction on the number of particles, as was proposed in our earlier manuscript (Chulaevsky and Suhov 2007). Recently, Aizenman and Warzel (2008) proved spectral and dynamical localisation for N-particle lattice systems with a short-range interaction, using an extension of the Fractional-Moment Method (FMM) developed earlier for single-particle models in Aizenman and Molchanov (Commun Math Phys 157:245–278, 1993) and Aizenman et al. (Commun Math Phys 224:219–253, 2001) (see also references therein) which is also combined with an induction on the number of particles.