In the thesis, we describe the dynamics of two many-body systems for Fermions.

The first system depicts infinitely many electrons moving in a constant magnetic field, where they interact with each other. In the reduced Hartree-Fock setting (ignoring the exchange term in the Hartree-Fock setting), the main part of electrons occupy the low energy state and are stationary, while the small part of them are excited particles. Based on this setting, we used Harmonic analysis and asymptotic properties of associated Laguerre polynomials to establish a local well-posedness theory, which is below the energy level.

The second system describes the motion of finitely many Fermions in the absence of background fields. In the Bogoliubov-de Gennes setting, based on the observation that the correlation function modeling Cooper pairs is anti-symmetric, we were able to employ techniques of dispersive equations and extend the existing global well-posedness theory. The well-posedness theory was proven for the Coulomb interaction potential. We extended it to the case with a more singular interaction potential $\frac{1}{|x|^{2-\epsilon}}$, for any $0\le \epsilon <2$.