Novel Quantum States of the Rational Calogero Models Without the Confining Interaction

TL;DR

This paper discovers new bound and scattering states in rational Calogero models without harmonic confinement, arising from self-adjoint extensions, and explores their properties and symmetry implications.

Contribution

It introduces a novel class of states in Calogero models without confining potential, linked to self-adjoint extensions, and generalizes findings to higher-dimensional systems.

Findings

Existence of new bound and scattering states for all N

States appear for specific coupling constant ranges

Self-adjoint extensions break classical scaling symmetry

Abstract

We show that the N-particle A_{N-1} and B_N rational Calogero models without the harmonic interaction admit a new class of bound and scattering states. These states owe their existence to the self-adjoint extensions of the corresponding Hamiltonians, labelled by a real parameter z. It is shown that the new states appear for all values of N and for specific ranges of the coupling constants. Moreover, they are shown to exist even in the excited sectors of the Calogero models. The self-adjoint extension generically breaks the classical scaling symmetry, leading to quantum mechanical scaling anomaly. The scaling symmetry can however be restored for certain values of the parameter z. We also generalize these results for many particle systems with classically scale invariant long range interactions in arbitrary dimensions.