New bounds on the excess charge for bosonic systems interacting through Coulomb potentials

Abstract

The following research results, whose main goal was to establish new upper bounds for the excess charge for an atom, are collected in this PhD thesis. Such an atom consists of a fixed, classical nucleus of charge $Z$ (Born-Oppenheimer approximation) and $N$ non-relativistic quantum particles (Hartree approximation). A technique inspired by the work of 1983 of R. Benguria and E. Lieb, combined with some elements of P. T. Nam's article of 2012, was applied and improved to achieve the purpose of this research. The improvement is the success in finding a coherent result for bosonic matter, $N(Z) \leq 1.4811Z+3.1516Z^{1/3}$ on the one hand, which is a better upper bound that the best known up-to-date established by E. Lieb for the same case $N(Z)<2Z+1$ on the other hand. A novel aspect of this joint work with R. Benguria is that instead of using Lieb-Thirring's inequalities, essentially fermionic ones, we use a different result, which prevents all $N$ particles from collapsing together into the nucleus: the Coulomb Uncertainty Principle.