New Non-Symmetric Orthogonal Basis for the Calogero Model with Distinguishable Particles

TL;DR

This paper introduces a new algebraic method to construct eigenfunctions for the Calogero model with distinguishable particles, revealing differences from existing polynomial approaches and expanding understanding of its algebraic structure.

Contribution

It presents a novel non-symmetric orthogonal basis construction for the Calogero model, distinct from previous polynomial-based methods, using an algebraic approach similar to quantum harmonic oscillators.

Findings

Constructed all eigenfunctions algebraically for the Calogero Hamiltonian with distinguishable particles.

Discovered the eigenfunctions differ from the non-symmetric Hi-Jack polynomial.

Showed the conserved operators are algebraically different from those derived via Dunkl operators.

Abstract

We demonstrate an algebraic construction of all the simultaneous eigenfunctions of the conserved operators for distinguishable particles governed by the Calogero Hamiltonian. Our construction is completely parallel to the construction of the Fock space for decoupled quantum harmonic oscillators. The simultaneous eigenfunction does not coincide with the non-symmetric Hi-Jack polynomial, which shows that the conserved operators derived from the number operators of the decoupled quantum harmonic oscillators are algebraically different from the known ones derived by the Dunkl operator formulation.