The DtN map naturally arises in physics, for example, in the studies of electrostatics and water waves. Heuristically, it sends the voltage measured at the boundary of a crystal to the density of current flowing through. In this talk, we discuss a new scattering theory for the DtN map on asymptotically conic domains with Lipschitz boundaries in any dimension, where the nonlocality of the DtN map has a non-trivial interaction with the singularity of the boundary. We then use the new scattering theory to fully identify the spectrum of the DtN map on 2D polygons with piecewise smooth boundaries, as an improvement to earlier work of Levitin–Parnovski–Polterovich–Sher. This is joint work with Jeff Galkowski and Marcello Malagutti.
