Hierarchical models with inverse-square interaction in harmonic confinement

A hierarchy of quantum models with inverse-square pairwise interaction is studied for the system with a confining harmonic potential. An approach based on renormalized harmonic oscillators is used to construct the energy spectrum, which naturally results in the renormalized Hartree-Fock description of the interaction. In the limit of weak confinement, the model may serve as a canonical model for multiple chiral Tomonaga-Luttinger liquids for the edge states of the fractional quantum Hall effect. For the SU() symmetric case, we find that the excitation spectrum is classified in terms of the simple roots and the weights of SU() Lie algebra.