Pseudo-Path Semiclassical Approximation to Transport through Open Quantum Billiards: Dyson Equation for Diffractive Scattering

We present a semiclassical theory for transport through open billiards of arbitrary convex shape that includes diffractively scattered paths at the lead openings. Starting from a Dyson equation for the semiclassical Green's function we develop a diagrammatic expansion that allows a systematic summation over classical paths and pseudo-paths which consist of classical paths joined by diffractive scatterings (``kinks''). This renders the inclusion of an exponentially proliferating number of pseudo-path combinations numerically tractable for both regular and chaotic billiards. For a circular billiard and the Bunimovich stadium the path sum leads to a good agreement with the quantum path length power spectrum up to long path length. Furthermore, we find excellent numerical agreement with experimental studies of quantum scattering in microwave billiards where pseudo-paths provide a significant contribution.