A great deal of progress has been made in recent years on the rigorous derivation of effective Non-Linear Schrödinger equations starting from N-body Schrödinger Hamiltonians. In particular, Bose-Einstein condensation of zero-temperature ground states of large bosonic systems has been derived for various models and asymptotic regimes.
In this talk we shall discuss a similar limit for positive temperature equilibrium states of the many-body Schrödinger Hamiltonian. We start from grand-canonical Gibbs states and obtain non-linear Gibbs measures built on the NLS functional in the limit. Our method covers the case of 1D particles with repulsive contact interactions, and higher dimensional particles with a reguralized interaction.
Joint work with Mathieu Lewin and Phan Thành Nam.