Spectral Properties of Point Interactions with Fermionic Symmetries

The history of quantum systems with zero-range interactions can be traced back to the 1930s, with the pioneering paper of Bethe and Peierls on the quantum theory of the diplon. The past two decades have witnessed a renewed interest in this class of models among both physicists and mathematicians. Advances in experimental techniques have made possible to produce systems of particles whose two-body range of interactions is virtually zero, leading to a class of phenomena yet to be understood in terms of basic principles of quantum mechanics. The main focus here is on particularly relevant spectral properties of these systems, such as their stability or the presence/absence of singular spectral phenomena (e.g., Efimov effect), in dimension three. In particular, our work gives an (almost) complete picture of the spectrum for two identical fermions interacting with a third particle of a different nature in a three dimensional space, provided that the latter has a sufficiently large mass compared to the fermions. Possible future developments are also discussed.