Small distance behaviour of one-particle Green’s functions in electronic structure theory

Within the framework of many-particle perturbation theory, we develop an analytical approach that allows us to determine the small distance behaviour of Green’s functions and related quantities in electronic structure theory. As a case study, we consider the one-particle Green’s function up to second-order in the perturbation approach. We derive explicit expressions for the leading order terms of the asymptotic small distance behaviour. In particular, we demonstrate the appearance of logarithmic terms in the corresponding second-order Feynman diagrams. Our asymptotic analysis leads to an improved classification scheme for the diagrams, which takes into account not only the perturbation order, but also the asymptotic smoothness properties near their diagonals. Such a classification may be useful in the design of numerical algorithms and helps to improve their efficiency.