Analytic results for N particles with 1/r2 interaction in two dimensions and an external magnetic field.

Abstract

The $2N$-dimensional quantum problem of $N$ particles (e.g., electrons) with interaction $\ensuremath{\beta}/{r}^{2}$ in a two-dimensional parabolic potential ${\ensuremath{\omega}}_{0}$ (e.g., quantum dot), and magnetic field $B$, reduces exactly to solving a $(2N\ensuremath{-}4)$-dimensional problem which is independent of $B$ and ${\ensuremath{\omega}}_{0}$. An exact, infinite set of relative mode excitations are obtained for any $N$. The $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3$ problem reduces to that of a fictitious particle in a two-dimensional, nonlinear potential of strength $\ensuremath{\beta}$, subject to a fictitious magnetic field ${B}_{\mathrm{fic}}\ensuremath{\propto}J$, the relative angular momentum.