Bose-Einstein Condensation in Presence of External Potentials

It is know that not only interaction but also external potentials are able to create or to change drastically the Bose-Einstein Condensation (BEC). I start my lectures by discussion of the Generalised Condensation (GC) a la van den Berg-Lewis-Pule, which is due to a pure geometrical constrains imposed on the bosons for thermodynamic limit in the Casimir boxes or in scaled external potentials. The notion of the GC is important in the next lectures concerning a condensation in random external potentials and in particular for analysis of the localised versus space extended condensation. I present also some results on the BEC in magnetic fields and about a specificity of condensation in weak harmonic traps. Essentially my lectures are focused on the properties of the equilibrium perfect (or mean-field) Bose-gas, although I present also some results for interacting bosons and for bosons in external potentials which put them out of equilibrium. (author)