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

This paper presents a systematic construction of exact Heisenberg‑operator solutions for a broad class of exactly solvable multi‑particle quantum systems, focusing on Calogero‑type models defined on arbitrary root systems. The authors begin by introducing the notion of “sinusoidal coordinates,” a set of (N) independent operators ({\eta_j}_{j=1}^N) that evolve in time exactly as simple harmonic functions. These coordinates are linear combinations of the particle positions, with coefficients determined by the geometry of the underlying root system, and they satisfy closed commutation relations with the Hamiltonian.

The Hamiltonian under study has the standard Calogero form
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