Computing singularities of perturbation series

Abstract

Many properties of current \emph{ab initio} approaches to the quantum many-body problem, both perturbational or otherwise, are related to the singularity structure of Rayleigh--Schr\"odinger perturbation theory. A numerical procedure is presented that in principle computes the complete set of singularities, including the dominant singularity which limits the radius of convergence. The method approximates the singularities as eigenvalues of a certain generalized eigenvalue equation which is solved using iterative techniques. It relies on computation of the action of the perturbed Hamiltonian on a vector, and does not rely on the terms in the perturbation series. Some illustrative model problems are studied, including a Helium-like model with $\delta$-function interactions for which M{\o}ller--Plesset perturbation theory is considered and the radius of convergence found.