A numerical method of analytical continuation of Green-function-like expressions

Green functions, being the basic entities of a great part of condensed matter theory, are 'Herglotz', i.e. they can be analytically continued into the open half-planes forming the physical sheet of complex energy. On the real-energy axis they are usually nonanalytic and may even be distributions, which makes their numerical calculation there hard to perform. On the contrary, being smooth functions in the complex plane, they may easily be computed numerically there, provided closed formal expressions are given. The problem of a numerical analytical continuation towards the singularities (deconvolution) has been treated rather empirically in recent literature. Here a systematic analysis of the extrapolation error propagation is given, and an optimised deconvolution procedure is proposed. The focus is on the spectral function, the density of states and integrated density of states problems. Examples are given demonstrating the power of the procedure.