Anderson localisation for an interacting two-particle quantum system on ${\mathbb Z}$

Abstract

We study spectral properties of a system of two quantum particles on an integer lattice with a bounded short-range two-body interaction, in an external random potential field $V(x,ω)$ with independent, identically distributed values. The main result is that if the common probability density $f$ of random variables $V(x,ω)$ is analytic in a strip around the real line and the amplitude constant $g$ is large enough (i.e. the system is at high disorder), then, with probability one, the spectrum of the two-particle lattice Schroedinger operator $H(ω)$ (bosonic or fermionic) is pure point, and all eigen-functions decay exponentially. The proof given in this paper is based on a refinement of a multiscale analysis (MSA) scheme proposed by von Dreifus and Klein, adapted to incorporate lattice systems with interaction.