Bound States in One-Dimensional Quantum N-Body Systems with Inverse Square Interaction

TL;DR

This paper explores the existence of bound states in a one-dimensional quantum N-body system with inverse square interactions, revealing that such states occur only for N=3,4 under specific conditions and are linked to the Virasoro algebra.

Contribution

It demonstrates the conditions under which bound states exist in the Calogero model without confinement, highlighting the role of boundary conditions and system parameters.

Findings

Bound states exist only for N=3,4.

Bound states depend on specific boundary conditions.

Wavefunctions show scaling behavior at small separations.

Abstract

We investigate the existence of bound states in a one-dimensional quantum system of $N$ identical particles interacting with each other through an inverse square potential. This system is equivalent to the Calogero model without the confining term. The effective Hamiltonian of this system in the radial direction admits a one-parameter family of self-adjoint extensions and the negative energy bound states occur when most general boundary conditions are considered. We find that these bound states exist only when $N=3,4$ and for certain values of the system parameters. The effective Hamiltonian for the system is related to the Virasoro algebra and the bound state wavefunctions exhibit a scaling behaviour in the limit of small inter-particle separation.