Petrat, Sören; Pickl, Peter
(2016):
A New Method and a New Scaling for Deriving Fermionic MeanField Dynamics.
In: Mathematical Physics Analysis and Geometry, Vol. 19, No. 1, 3

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Abstract
We introduce a new method for deriving the timedependent Hartree or HartreeFock equations as an effective meanfield dynamics from the microscopic Schrodinger equation for fermionic manyparticle systems in quantum mechanics. The method is an adaption of the method used in Pickl (Lett. Math. Phys. 97 (2) 151164 2011) for bosonic systems to fermionic systems. It is based on a Gronwall type estimate for a suitable measure of distance between the microscopic solution and an antisymmetrized product state. We use this method to treat a new meanfield limit for fermions with longrange interactions in a large volume. Some of our results hold for singular attractive or repulsive interactions. We can also treat Coulomb interaction assuming either a mild singularity cutoff or certain regularity conditions on the solutions to the Hartree(Fock) equations. In the considered limit, the kinetic and interaction energy are of the same order, while the average force is subleading. For some interactions, we prove that the Hartree(Fock) dynamics is a more accurate approximation than a simpler dynamics that one would expect from the subleading force. With our method we also treat the meanfield limit coupled to a semiclassical limit, which was discussed in the literature before, and we recover some of the previous results. All results hold for initial data close (but not necessarily equal) to antisymmetrized product states and we always provide explicit rates of convergence.