Consider a system of N quantum particles; their dynamics is described by the Schroedinger equation with potentials which are possibly singular, long range and time-dependent. Mathematical Scattering Theory studies the large time behaviour of the solutions and determines all possible configurations in the remote past and in the far future (freely moving particles and composites, atoms and molecules). In this talk we illustrate the mathematical structure of the theory in the simplest N=2 case.

*profile:* Kenji Yajima (PhD in Mathematics 1978, University of Tokyo), a prominent member of the Japan school of analysis and mathematical physics, has been a professor at Princeton University and at the University of Tokyo, as well as president of the Mathematical Society of Japan. He is now professor at the university of Tokyo Gakushuin. He is a renowned expert in partial differential equations and scattering theory. He gave seminal contributions on the existence, uniqueness, and regularity for the initial value problem of the Schrödinger equation and on the W^{k,p}-continuity of its wave operators (the "Yajima bounds"). He has an ample score of long-term visiting professorships all around the world, including ETH Zurich, University of Virginia, Princeton University, Courant Institute, University of Vienna, Johns-Hopkins University, University of Rome La Sapienza, and (in April-July and October-November 2016) at SISSA Trieste.