An elementary construction of lowering and raising operators for the trigonometric Calogero-Sutherland model

TL;DR

This paper presents an elementary method to construct lowering and raising operators for the trigonometric Calogero-Sutherland model, extending previous rational case techniques to the more complex trigonometric scenario.

Contribution

It provides a novel, elementary construction of these operators specifically for the trigonometric Calogero-Sutherland model, addressing the challenges posed by its non-equidistant spectrum.

Findings

Constructed explicit lowering and raising operators for the trigonometric model.

Extended the algebraic framework from rational to trigonometric case.

Demonstrated the operators' effectiveness in the non-equidistant spectrum context.

Abstract

Quantum Calogero-Sutherland model of $A_n$ type is completely integrable. Using this fact, we give an elementary construction of lowering an raising operators for the trigonometric case. This is similar, but more complicated (due to the fact that the energy spectrum is not equidistant) than the construction for the rational case.