Inequivalent Quantizations of the Rational Calogero Model

TL;DR

This paper demonstrates that the rational Calogero model admits multiple inequivalent quantizations through different self-adjoint extensions, resulting in diverse spectra including non-equispaced energy levels with both positive and negative energies.

Contribution

It introduces the concept of inequivalent quantizations of the rational Calogero model via boundary conditions, expanding understanding of its spectral properties.

Findings

Existence of non-equispaced energy levels in the spectrum.

Presence of a negative energy state for certain boundary conditions.

Multiple spectra corresponding to different self-adjoint extensions.

Abstract

We show that the rational Calogero model with suitable boundary condition admits quantum states with non-equispaced energy levels. Such a spectrum generically consists of infinitely many positive energy states and a single negative energy state. The new states appear for arbitrary number of particles and for specific ranges of the coupling constant. These states owe their existence to the self-adjoint extensions of the corresponding Hamiltonian, which are labelled by a real parameter z. Each value of z corresponds to a particular spectrum, leading to inequivalent quantizations of the model.