Mathematics

Bose-Einstein condensation with multiple components represents one of the current frontiers in cold atom experiments; their mathematical treatment is a recently developing and already quite active field, involving operator-theoretic and functional-analytic techniques combined with scaling limits. After a (partial) overview of the rigorous description of BEC, I will introduce two major types of multi-component condensates, mixtures and pseudo-spinors. For them, the effective dynamics is well described by suitable systems of coupled non-linear Schroedinger equations, the rigorous derivation of which from the many-body linear Schroedinger dynamics was proved first in recent joint works with Alessandro Michelangeli. I will also discuss the yet different case of the so-called fragmented condensates, whose effective dynamics is missing so far a rigorous mathematical treatment.