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

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Abstract: We discuss a basis set developed to calculate perturbation coefficients in anexpansion of the general N-body problem. This basis has two advantages. First,the basis is complete order-by-order for the perturbation series. Second, thenumber of independent basis tensors spanning the space for a given order doesnot scale with N, the number of particles, despite the generality of theproblem. At first order, the number of basis tensors is 23 for all N althoughthe problem at first order scales as N^6. The perturbation series is expandedin inverse powers of the spatial dimension. This results in a maximallysymmetric configuration at lowest order which has a point group isomorphic withthe symmetric group, S N. The resulting perturbation series is order-by-orderinvariant under the N! operations of the S N point group which is responsiblefor the slower than exponential growth of the basis. In this paper, we performthe first test of this formalism including the completeness of the basisthrough first order by comparing to an exactly solvable fully-interactingproblem of N particles with a two-body harmonic interaction potential.

Author: ** W. Blake Laing, David W. Kelle, Martin Dunn, Deborah K. Watson**

Source: https://arxiv.org/