Derivation of the Landau-Pekar Equations in a Many-Body Mean-Field Limit.

We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle numbers and a suitably small coupling, we show that the dynamics of the system is approximately described by the Landau-Pekar equations. These describe a Bose-Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.

Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions.

We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross-Pitaevskii equation starting from an interacting -particle system of bosons. We consider the interaction potential to be given either by , for any , or to be given by , for some spherical symmetric, nonnegative and compactly supported . In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm.