Analytic results for $N$ particles with $1/r^2$ interaction in two dimensions and an external magnetic field

TL;DR

This paper derives exact solutions for a system of N particles with inverse-square interactions in two dimensions under a magnetic field, revealing a reduction to a lower-dimensional problem and providing a set of relative mode excitations.

Contribution

It presents an exact reduction of the N-particle quantum problem with 1/r^2 interaction in 2D and magnetic field to a lower-dimensional problem, and finds an infinite set of relative mode excitations.

Findings

Exact reduction of the N-particle problem to a (2N-4)-dimensional problem

Derivation of an infinite set of relative mode excitations

Reduction of the N=3 problem to a fictitious particle in a non-linear potential

Abstract

The $2N$-dimensional quantum problem of $N$ particles (e.g. electrons) with interaction $\beta/r^2$ in a two-dimensional parabolic potential $\omega_0$ (e.g. quantum dot) and magnetic field $B$, reduces exactly to solving a $(2N-4)$-dimensional problem which is independent of $B$ and $\omega_0$. An exact, infinite set of relative mode excitations are obtained for any $N$. The $N=3$ problem reduces to that of a ficticious particle in a two-dimensional, non-linear potential of strength $\beta$, subject to a ficticious magnetic field $B_{\rm fic}\propto J$, the relative angular momentum.