Computing singularities of perturbation series

TL;DR

This paper introduces a numerical method to identify all singularities in Rayleigh--Schr"odinger perturbation theory, including the dominant one that determines the convergence limit, by solving a generalized eigenvalue problem.

Contribution

It presents a novel iterative approach to compute the complete singularity structure of perturbation series without relying on series terms.

Findings

Successfully applied to model problems including Helium-like systems

Identified the dominant singularity limiting convergence

Demonstrated effectiveness with alculus-based eigenvalue computations

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.