Computing singularities of perturbation series

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.

💡 Research Summary

The paper addresses a fundamental limitation of Rayleigh‑Schrödinger perturbation theory in quantum many‑body calculations: the radius of convergence is dictated by the nearest singularity of the energy as a function of the complex perturbation parameter λ. Traditional approaches to locate these singularities rely on explicit high‑order coefficients, Padé approximants, or analytic continuation, all of which become prohibitively expensive for realistic electronic‑structure problems.

The authors propose a radically different strategy. They reformulate the problem of finding singularities as a generalized eigenvalue problem
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