Analytic-continuation approach to the resummation of divergent series in Rayleigh-Schrödinger perturbation theory

Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them

“…In a later study by the same group, they used analytic continuation techniques to resum a divergent MP series such as a stretched water molecule. 59 Any MP series truncated at a given order n can be used to define the scaled function…”

“…Although these EPs are generally complex-valued, their positions are intimately related to the convergence of the perturbation expansion on the real axis. [55][56][57][58][59][60][61] Furthermore, the existence of an avoided crossing on the real axis is indicative of a nearby EP in the complex plane. Our aim in this article is to provide a comprehensive review of the fundamental relationship between EPs and the convergence properties of the MP series.…”

The inverse boundary value problem: application in many-body perturbation theory

“…In Ref. [1], we emphasized that values E(z) can be obtained at no further cost once the PT contributions E (n) are available, by scaling the nth-order term by z n a posteriori (cf. Eq.…”

“…In a recent paper [1], we introduced an a posteriori scaling of individual terms E (n) of a series which for z = 1 recovers the original (eventually divergent) series, but for |z| < |z 0 | produces a convergent one if |z 0 | is the radius of convergence. E (0) is identified as the eigenvalue of H 0 , and E (n) denotes the nth-order contribution to E(z = 1) .…”

“…The idea followed in Ref. [1] was based on function E(z) being analytic along a path from a point z (with |z| < |z 0 |) to the point z = 1 . In such a case, analytic continuation techniques can be used to extrapolate the values E(z) in the "trusted region" (i.e., for |z| < |z 0 | ) to z = 1 which is the point of physical interest.…”