Duality and interacting families in models with the inverse-squared interaction

TL;DR

This paper explores duality relations in a quantum many-body system with inverse-squared interactions, revealing algebraic solutions and hierarchical structures due to underlying symmetries.

Contribution

It demonstrates the presence of weak-strong coupling duality in inverse-squared interaction models and provides algebraic solutions using O(2,1) symmetry.

Findings

Duality relates coupling constants as inverses.

Spectrum and eigenfunctions are obtained algebraically.

Hierarchical Hamiltonian structure is established.

Abstract

Weak-strong coupling duality relations are shown to be present in the quantum-mechanical many-body system with the interacting potential proportional to the pair-wise inverse-squared distance in addition to the harmonic potential. Using duality relations we have solved the problem of families interacting by the inverse-squared interaction. Owing to duality, the coupling constants of the families are mutually inverse. The spectrum and eigenfunctions are determined mainly algebraically owing to O(2,1) dynamical symmetry. The constructed Hamiltonian for families and appropriate solutions are of hierarchical nature.