Exact Heisenberg operator solutions for multi-particle quantum mechanics

TL;DR

This paper derives exact Heisenberg operator solutions for multi-particle quantum systems, specifically Calogero models, providing explicit annihilation-creation operators and generalizing recursion relations for multi-variable orthogonal polynomials.

Contribution

It presents the first exact Heisenberg operator solutions for multi-particle Calogero systems with sinusoidal coordinates, extending the understanding of their algebraic structure.

Findings

Explicit Heisenberg solutions for Calogero systems derived

Annihilation-creation operators explicitly constructed

Generalization of recursion relations for multi-variable orthogonal polynomials

Abstract

Exact Heisenberg operator solutions for independent `sinusoidal coordinates' as many as the degree of freedom are derived for typical exactly solvable multi-particle quantum mechanical systems, the Calogero systems based on any root system. These Heisenberg operator solutions also present the explicit forms of the annihilation-creation operators for various quanta in the interacting multi-particle systems. At the same time they can be interpreted as multi-variable generalisation of the three term recursion relations for multi-variable orthogonal polynomials constituting the eigenfunctions.