The low density Bose gas. (Lecturer: Prof. R. Seiringer)

The experimental observation of Bose-Einstein Condensation (BEC) in trapped gases of bosonic cold atoms has revived the interest of theoreticians in the problem of understanding BEC and related phenomena (superfluidity and vortex formation) in ultracold quantum gases. When the unavoidable inter-atomic interactions are taken into account this is a formidable task and despite remarkable progress in recent years some of the most important basic problems like the proof of BEC in the thermodynamic limit, starting from a many-body Hamiltonian with realistic interactions, are still unsolved. Hence this area offers many challenges for mathematical physics. In this course the focus will be on dilute Bose gases in magneto-optical traps. The passage from the ground state of the full many-body Hamiltonian with repulsive interactions to an effective description in terms of the single-particle Gross-Pitaevskii equation will be discussed in some detail. In particular we shall prove that BEC takes place in the Gross-Pitaevskii limit for a trapped low density gas of bosonic atoms and show how the ground state solution of the Gross-Pitaevskii equation emerges as the macroscopic wave function of the Bose-Einstein condensate. The Gross-Pitaevskii limit is a limit where the ratio of the interaction energy per particle to the spectral gap in the trap is kept constant as the particle number tends to infinity. Although this is a weaker result than a proof of BEC in the thermodynamic limit it is of great importance for the interpretation of current experiments with trapped gases. The proof is based on upper and lower bounds on the ground state kinetic and potential energy, obtained via variational bounds and the combination of a number of remarkable functional inequalities that will be discussed in the lectures. The technique for obtaining an asymptotically correct lower bound to the ground state energy has been a key to the solution of several other problems like the dimensional reduction of Bose gases in highly elongated (``cigar shaped'') or thin, ``disc shaped'' traps. If time permits, extensions to the Bose gases in rotating containers will be discussed also. In this case new phenomena, that are related to the creation of vortices in the rotating gas, occur. In particular, the ground state needs not be unique since the vortex pattern can break rotational symmetry. The derivation of the Gross-Pitaevskii equation and the proof of BEC requires in this case different methods from the non-rotating situation. An important ingredient of the proof of the lower bound to the ground state energy is the use of coherent states, a technique that has a wide range of other applications.