The Rayleigh-Schr\"odinger perturbation series of quasi-degenerate systems

TL;DR

This paper introduces a novel tree-based representation for the Rayleigh-Schr"odinger perturbation series in quasi-degenerate systems, enabling new combinatorial and resummation techniques.

Contribution

It provides the first general term representation for the series in quasi-degenerate systems using trees, linking combinatorial objects with analytical expressions.

Findings

Tree representation of perturbation series terms

Derivation of resummation formulas

Connection with combinatorial objects for special cases

Abstract

We present the first representation of the general term of the Rayleigh-Schr\"odinger series for quasidegenerate systems. Each term of the series is represented by a tree and there is a straightforward relation between the tree and the analytical expression of the corresponding term. The combinatorial and graphical techniques used in the proof of the series expansion allow us to derive various resummation formulas of the series. The relation with several combinatorial objects used for special cases (degenerate or non-degenerate systems) is established.