Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes

We give a self-contained description of the main results from the paper (Cacciapuoti et al., J Math Anal Appl 473(1):215–257, 2019). We focus on the fundamental concepts and on the chief achievements, omitting some auxiliary results and a number of technical details given in the original paper. We discuss the scattering problem for a quantum particle in dimension three in the presence of a semitransparent unbounded obstacle, modelled by a δ-interaction supported on a surface obtained through a local, Lipschitz continuous deformation of a flat plane. We discuss existence and asymptotic completeness of the wave operators with respect to a suitable reference dynamics. Additionally, we provide an explicit expression for the related scattering matrix and show that it converges to the identity as the deformation goes to zero (giving a quantitative estimates on the rate of convergence).