We present a microscopic derivation of the time-dependent Gross-Pitaevskii equation starting from an interacting N -particle system of Bosons. We prove convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the respective Gross-Pitaevskii equation. Our work extends a previous result by one of us (P.P.[44]) to interaction potentials which need not to be nonnegative, but may have a sufficiently small negative part.

We consider the Maxwell-Schrödinger equations in the Coulomb gauge describing the interaction of extended fermions with their self-generated electromagnetic field. They heuristically emerge as mean-field equations from non-relativistic quantum electrodynamics in a mean-field limit of many fermions. In the semiclassical regime, we establish the convergence of the Maxwell-Schrödinger equations for extended charges towards the nonrelativistic Vlasov-Maxwell dynamics and provide explicit estimates on the accuracy of the approximation. To this end, we build a well-posedness and regularity theory for the Maxwell-Schrödinger equations and for the Vlasov-Maxwell system for extended charges.

We report on a simple strategy to treat mean-field limits of quantum mechanical systems in which a large number of particles weakly couple to a second-quantized radiation field. Extending the method of counting, introduced in [21], with ideas inspired by [16] and [6] leads to a technique that can be seen as a combination of the method of counting and the coherent state approach. It is similar to the coherent state approach but might be slightly better suited to systems in which a fixed number of particles couple to radiation. The strategy is effective and provides explicit error bounds. As an instructional example we derive the Schrödinger-Klein-Gordon system of equations from the Nelson model with ultraviolet cutoff. Furthermore, we derive explicit bounds on the rate of convergence of the one-particle reduced density matrix of the non-relativistic particles in Sobolev norm. More complicated models like the Pauli-Fierz Hamiltonian can be treated in a similar manner [14].MSC class: 35Q40, 81Q05, 82C10

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.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order.

The Nelson model (with ultraviolet cutoff) describes a quantum system of non-relativistic particles coupled to a positive or zero mass quantized scalar field. We take the nonrelativistic particles to obey Fermi statistics and discuss the time evolution in a mean-field limit of many fermions which is coupled to a semiclassical limit. At time zero, we assume that the bosons of the radiation field are close to a coherent state and that the state of the fermions is close to a Slater determinant with a certain semiclassical structure. We show that the many-body state approximately stays a Slater determinant and retains its semiclassical structure at later times and that its time evolution can be approximated by the fermionic Schrödinger-Klein-Gordon equations. This is proven in terms of reduced density matrices with explicit rates of convergence and for all semiclassical times.MSC class: 35Q55, 81Q05, 81T10, 82C10