A Complete Basis for a Perturbation Expansion of the General N-Body Problem

TL;DR

This paper introduces a complete and efficient basis set for perturbation expansion in the general N-body problem, maintaining manageable complexity regardless of particle number, and validates it against an exactly solvable model.

Contribution

The authors develop a basis set that is complete order-by-order and independent of N, enabling scalable perturbation analysis of the N-body problem.

Findings

Basis set is complete order-by-order for all N

Number of basis tensors at first order is 23 regardless of N

Perturbation series invariant under symmetric group operations

Abstract

We discuss a basis set developed to calculate perturbation coefficients in an expansion of the general N-body problem. This basis has two advantages. First, the basis is complete order-by-order for the perturbation series. Second, the number of independent basis tensors spanning the space for a given order does not scale with N, the number of particles, despite the generality of the problem. At first order, the number of basis tensors is 23 for all N although the problem at first order scales as N^6. The perturbation series is expanded in inverse powers of the spatial dimension. This results in a maximally symmetric configuration at lowest order which has a point group isomorphic with the symmetric group, S_N. The resulting perturbation series is order-by-order invariant under the N! operations of the S_N point group which is responsible for the slower than exponential growth of the basis. In this paper, we perform the first test of this formalism including the completeness of the basis through first order by comparing to an exactly solvable fully-interacting problem of N particles with a two-body harmonic interaction potential.