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Applied Math Seminar
Anderson Localisation for Multi-Particle Systems
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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) |