The X-ray problem revisited

In this letter we re-visit the X-ray problem. Assuming point interaction between the conduction electrons and the instantaneously created core-hole, the latter’s Green’s function can be represented as a Fredholm determinant of certain Wiener-Hopf operators acting on L2(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 spend 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 asymptotics. We consider the classical X-ray problem where the core-hole is created at the origin at t = 0 and removed at t = T > 0. The object here is to study the behaviour of the corehole Green’s function as T gets large. By first integrating out the core-hole followed by the conduction electrons [4] or using a diagrammatic approach [15] the core-hole Green’s reads, G(T ) G(0)(T ) = det(A− vB) detA , (1) where v > 0, is the strength of local potential and G is the free core-hole Green’s function. ∗Supported in part by NSF Grants DMS-9623278 and DMS-9970879.