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Effective dynamics of interacting fermionic systems

Speaker: 
Marcello Porta
Institution: 
University of Zurich and University of Tubingen
Schedule: 
Monday, September 25, 2017 - 14:30
Location: 
A-136
Abstract: 

The dynamics of a quantum system is described by the Schroedinger equation, a linear evolution equation. For realistic values of the number of particles, however, it is essentially impossible to extract information from its solution. For this reason, physicists introduced effective evolution equations, much simpler to study than the full Schroedinger evolution, that are expected to provide a good approximation of the dynamics in suitable scaling regimes. In this talk, I will discuss the evolution of interacting fermionic systems, in the mean-field regime. I will consider initial data which are close to Slater determinants, describing initially confined zero temperature states, with a suitable semiclassical structure. For bounded interaction potentials, I will show that the many-body evolution converges to the solution of the time-dependent Hartree-Fock equation, a celebrated nonlinear effective equation. I will then discuss the extension to Coulomb potentials, under the assumption that the solution of the time-dependent Hartree-Fock equation preserves the semiclassical structure of the initial datum. The results hold for all times, and give effective bounds on the rate of convergence from the many-body quantum dynamics to the Hartree-Fock dynamics as the number of particles goes to infinity.

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