We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schroedinger equation for fermionic many-particle systems in quantum mechanics. The method is an adaption of the method used in [Pickl, Lett. Math. Phys., 97(2):151-164, 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 mean-field limit for fermions with long-range 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 mean-field 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.
Ground States of Nonlinear Fermionic Systems: From Power-Law Interactions to Logarithmic Sobolev Inequality
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Central China Normal University（华中师范大学）；School of Mathematics and Information Science, Zhongyuan University of Technology（中原工学院数学与信息科学学院）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学国家重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
