Analytic Results for $\mathit{N}$ Particles with $1/{\mathit{r}}^2$ Interaction in Two Dimensions and an External Magnetic Field

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 $N=3\mathrm{}$ problem reduces to that of a fictitious particle in a two-dimensional, nonlinear potential of strength $\beta$, subject to a fictitious magnetic field $B_{\mathrm{fic}}\propto J$, the relative angular momentum.