The X-ray problem revisited

Abstract

In this letter we re-visit the X-ray problem. Assuming point interaction between the conduction electrons and the first instantaneously created core-hole, the latter's Green's function can be represented as a Fredholm determinant of certain Wiener-Hopf operators acting on L^2(0,T) with discontinuous symbols. Here the symbols are the local conduction electron Green's function in the frequency domain and T is the time the core-hole spends in the system before removal. In this situation, the classical theory of singular integral equations usually employed in the literature to compute the large T asymptotics of the Fredholm determinant ceased to be applicable. A rigorous theory first put forward in the context of operator theory comes into play and universal constants are found in the aymptotics.