Intertwining Relations for the Matrix Calogero-like Models: Supersymmetry and Shape Invariance

TL;DR

This paper develops new intertwining relations for matrix Calogero-like models, introducing local differential operators, and constructs exactly solvable Dirac-like equations and shape-invariant matrix models, linking to supersymmetric quantum mechanics.

Contribution

It introduces a novel class of local intertwining operators for matrix Calogero-like models and connects these to supersymmetry and shape invariance in quantum mechanics.

Findings

Constructed new local differential operators intertwining matrix Hamiltonians.

Developed exactly solvable Dirac-like equations within the models.

Established a connection with multidimensional supersymmetric quantum mechanics.

Abstract

Intertwining relations for $N$-particle Calogero-like models with internal degrees of freedom are investigated. Starting from the well known Dunkl-Polychronakos operators, we construct new kind of local (without exchange operation) differential operators. These operators intertwine the matrix Hamiltonians corresponding to irreducible representations of the permutation group $S_N$. In particular cases, this method allows to construct a new class of exactly solvable Dirac-like equations and a new class of matrix models with shape invariance. The connection with approach of multidimensional supersymmetric quantum mechanics is established.