This work is an up-to-date and partially improved analysis of the basics of the mathe-
matical description of Bose-Einstein condensation in terms of first principles of Quantum
Mechanics.
The main aims are:
1) to provide a compact, yet coherent overview:
- of the basic mathematical tools used to formalise Bose-Einstein condensation,
- of the mathematical techniques for studying several features of this physical phenomenon,
- and of how such means emerge as the natural ones in connection with their physical interpretation;
2) to discuss and to place in the above perspective some new contributions and improvements:
- on equivalent characterizations of Bose-Einstein condensation,
- on the strength of the convergence for some currently available asymptotic results,
- and on the effects of interparticle correlations on the energy and the dynamics of the many-body condensate.
