Scattering at threshold in massive wave propagation and ionization

Abstract

In this thesis, we examine the transition between Legendrian regularity and ellipticity at infinity for two PDEs, the Schrödinger–Helmholtz equation with an attractive long range potential and the Klein–Gordon equation. In the former case, the transition occurs as the spectral parameter 𝐸 → 0⁺, and thus describes the ionization of Hydrogenic atoms, and in the latter case the transition occurs at null infinity, where the phase in the asymptotic tails becomes singular. Using novel microlocal tools, we work in the variable coefficient setting throughout.