Bose-Einstein condensation in external potentials

Bose-Einstein condensation (BEC) of the free boson gas was predicted in 1925 and rehabilitated only in 1938 by F. London, who was motivated by superfluidity of the liquid Helium-4. BEC is a subtle collective quantum phenomenon which can be mathematically expressed as formation of a coherent state (vector) in the boson Fock space. Recent mathematical studies showed that the structure of the BEC may be more complicated (generalized BEC a la van den Berg-Lewis-Pule or a dynamical condensation) than the one predicted in 1925. This concerns the free Bose gas, as well as the systems with particle interactions or embedded in external potentials, like in the recent experiments with bosons in traps (Nobel Prize 2001) and an important progress in the mathematical description of these systems by Lieb-Seiringer-Solovej-Yngvason. In spite of that the BEC is still a challenging problem of Mathematical Physics. In my lecture I am going to discuss mathematical problems related to only one particular question: how can BEC be modified by external potentials? I give a review of some cases in which one can prove that it survives (and even amplifies), including the cases of traps, random potentials and magnetic/electric fields.