Calogero-Moser System

Calogero-Moser systems are integrable systems with singular pair interactions. The rational Calogero-Moser system is the Hamiltonian system defined by

(1)

where , the coordinates are distinct real numbers, the momenta are real, and . Hamilton's equations give

			(2)
			(3)

Its Lax pair can be written using matrices and with entries

			(4)
			(5)
			(6)
			(7)

where the off-diagonal formulas apply for and is the imaginary unit. The evolution equation is

(8)

so the eigenvalues of are conserved. The matrix traces

(9)

for , ..., are integrals of motion that commute under the Poisson bracket and are functionally independent on a dense open set. In particular, (Moser 1975; Etingof 2007, pp. 11-19).

Adding the harmonic term , where , gives equilibrium coordinates

(10)

where the are the zeros of the Hermite polynomial with leading coefficient (Odake and Sasaki 2002). The quantum rational model with harmonic confinement is a Calogero-Sutherland model.

The equilibrium coordinates for and , 5, and 8 are illustrated above. The blue curves are the Hermite polynomials multiplied by and scaled independently for display. This positive factor preserves their zeros, which are marked as the orange equilibrium positions.