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

Abstract

The $2N$-dimensional quantum problem of $N$ particles (e.g. electrons) with interaction $β/r^2$ in a two-dimensional parabolic potential $ω_0$ (e.g. quantum dot) and magnetic field $B$, reduces exactly to solving a $(2N-4)$-dimensional problem which is independent of $B$ and $ω_0$. 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 $β$, subject to a ficticious magnetic field $B_{\rm fic}\propto J$, the relative angular momentum.