Abstract: A central topic in mathematical physics is the investigation of quantum systems subject to very short range potentials, virtually supported on a nite set of points. In this talk, after a preliminary recall of the denition and the main features of Schrödinger operators with point interactions, I will discuss some recent results on their scattering and smoothing properties. I will also discuss the characterization of the Sobolev norms induced by such operators.
Based on joint works with G. Dell'Antonio, V. Georgiev, F. Iandoli, A. Michelangeli, and K.Yajima.