Abstract: 

It is known that the stationary two-body Schr\”odinger equation can be realized as a pair of Fredholm maps using techniques from microlocal analysis, leading to a refined understanding of the resolvents. In this talk, we will review this theory and describe the modifications required to extend it to the three-body setting. In doing so, we will see that a further procedure, known as `second microlocalization’, becomes necessary. Such a procedure has already appeared in the two-body case near zero-energy. We will explain how second microlocalization can be carried out in the three-body case, and, if time permits, discuss how microlocal regularity can be propagated within the second microlocalized framework.