EAN: 9783319248967

EFFECTIVE EVOLUTION EQUATIONS FROM QUANTUM DYNA...

These notes investigate the time evolution of quantum systems  and in particular the rigorous
derivation of effective equations approximating the many-body Schrödinger dynamics in certain
physically interesting regimes. The focus is primarily on the derivation of time-dependent
effective theories (non-equilibrium question) approximating many-body quantum dynamics. The
book is divided into seven sections  the first of which briefly reviews the main properties of
many-body quantum systems and their time evolution. Section 2 introduces the mean-field regime
for bosonic systems and explains how the many-body dynamics can be approximated in this limit
using the Hartree equation. Section 3 presents a method  based on the use of coherent states 
for rigorously proving the convergence towards the Hartree dynamics  while the fluctuations
around the Hartree equation are considered in Section 4. Section 5 focuses on a discussion of a
more subtle regime  in which the many-body evolution can be approximated by means of the
nonlinear Gross-Pitaevskii equation. Section 6 addresses fermionic systems (characterized by
antisymmetric wave functions)  here  the fermionic mean-field regime is naturally linked with a
semiclassical regime  and it is proven that the evolution of approximate Slater determinants
can be approximated using the nonlinear Hartree-Fock equation. In closing  Section 7 reexamines
the same fermionic mean-field regime  but with a focus on mixed quasi-free initial data
approximating thermal states at positive temperature.

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