Cambridge

Anderson localisation is an important phenomenon describing a

transition between insulation and conductivity. The problem is to analyse

the spectrum of a Schroedinger operator with a random potential in the

Euclidean space or on a lattice. We say that the system exhibits

(exponential) localisation if with probability one the spectrum is pure

point and the corresponding eigen-functions decay exponentially fast.

So far in the literature one considered a single-particle model where the

potential at different sites is IID or has a controlled decay of

correlations. The present talk aims at $N$-particle systems (bosons or

fermions) where the potential sums over different sites, and the traditional

approach needs serious modifications. The main result is that if the

`randomness' is strong enough, the $N$-particle system exhibits

localisation.

The proof exploits the muli-scale analysis scheme going back to Froehlich,

Martinelli, Scoppola and Spencer and refined by von Drefus and Klein. No

preliminary knowledge of the related material will be assumed from the

audience, apart from basic facts.

This is a joint work with V Chulaevsky (University of Reims, France)