Equivalence of the super Lax and local Dunkl operators for Calogero-like models

TL;DR

This paper constructs super Lax operators for Calogero models, linking them to local Dunkl operators, and demonstrates their equivalence, facilitating the derivation of eigenfunctions, integrals of motion, and relations between models.

Contribution

It introduces a novel construction of super Lax operators for Calogero models and establishes their equivalence with local Dunkl operators, simplifying analysis of these models.

Findings

Super Lax operators for Calogero models are constructed.

Super Lax operators are expressed in terms of supercharges and Dunkl operators.

Key relations between Lax matrices, Hamiltonians, and Dunkl operators are derived.

Abstract

Following Shastry and Sutherland I construct the super Lax operators for the Calogero model in the oscillator potential. These operators can be used for the derivation of the eigenfunctions and integrals of motion of the Calogero model and its supersymmetric version. They allow to infer several relations involving the Lax matrices for this model in a fast way. It is shown that the super Lax operators for the Calogero and Sutherland models can be expressed in terms of the supercharges and so called local Dunkl operators constructed in our recent paper with M. Ioffe. Several important relations involving Lax matrices and Hamiltonians of the Calogero and Sutherland models are easily derived from the properties of Dunkl operators.