This research reviewed in this habilitation thesis touches upon a variety of topics, grouped into six separate chapters. The unifying theme of all this work is a connection to the theory of many-body quantum systems. In some cases, the mathematical results are of direct relevance to such systems, like $N$-body systems or particle-field systems. In others, the connection lies in the fact that the studied equations are used as effective descriptions of many-body systems, for example in the mean-field approximation.