CalogeroMoser

Calogero-Moser refers to a family of exactly solvable classical and quantum integrable systems of N particles on a line with long-range pairwise interactions inversely proportional to the square of their separation. The rational version was introduced by Francesco Calogero in 1969 and later given a Lax formulation by Jürgen Moser in 1975, which established its complete integrability.

In the simplest rational form, the Hamiltonian is H = 1/2 sum_{i=1}^N p_i^2 + g sum_{i<j} 1/(x_i - x_j)^2. A confining harmonic term yields H = 1/2 sum p_i^2 + (ω^2/2) sum x_i^2 + g sum_{i<j} 1/(x_i - x_j)^2. Variants replace the rational interaction with trigonometric or elliptic ones, giving models with interactions 1/sin^2 or with the Weierstrass elliptic function, respectively. These are commonly called the trigonometric and elliptic Calogero-Moser models.

The models are Liouville integrable and possess a Lax pair (L, M) whose isospectral evolution generates a complete set of conserved quantities in involution. The Lax matrix for the rational case can be written with L_ij = p_i δ_ij + g (1-δ_ij)/(x_i - x_j).

Quantum Calogero models share many features with the classical case; the spectrum and eigenfunctions can be described using Dunkl operators and symmetric polynomials (e.g., Jack polynomials in certain cases). The system has deep connections to representation theory, algebraic geometry (Calogero-Moser spaces), and to integrable spin chains via the freezing trick. A relativistic generalization exists (Ruijsenaars–Schneider), and the elliptic version ties to rich algebraic structures.