Exact Heisenberg operator solutions for multi-particle quantum mechanics

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.

💡 Research Summary

In this paper the authors present a comprehensive construction of exact Heisenberg‑operator solutions for a broad class of exactly solvable multi‑particle quantum systems, namely the Calogero‑Sutherland‑Moser models associated with arbitrary root systems Δ. The work builds on the authors’ earlier studies of single‑degree‑of‑freedom systems and extends the methodology to r‑dimensional interacting particle systems, where r is the rank of the underlying root system.

The starting point is the quantum Calogero Hamiltonian \