Calogero-Sutherland Model

A Calogero-Sutherland model is an exactly solvable many-variable eigenvalue problem for a Schrödinger operator with singular pair interactions. The rational model with harmonic confinement has quantum Hamiltonian

(1)

where and . For on the region , choose the collision behavior proportional to . The unnormalized ground state and its eigenvalue are

			(2)
			(3)

Put and . A family of eigenfunctions is

(4)

where is a Hermite polynomial, is an associated Laguerre polynomial, , and and are nonnegative integers. These satisfy

(5)

Illustrated above are the signed eigenfunctions for , , , , and , 1, and 2, restricted to and scaled independently for display. The relative coordinates are

			(6)
			(7)

so . Blue and orange indicate positive and negative values. The three collision lines and are zeros of the common prefactor, and the radial zeros of the associated Laguerre polynomial give the nodal circles.

The case is the Hermite-Laguerre example in Trott (2006), following Song and Park (2002). More general eigenfunctions have multivariate Hermite polynomials as polynomial factors (Gonera et al. 1999, Desrosiers et al. 2003).

The trigonometric model on a circle is also called the Sutherland model. In angular coordinates, one normalization is

(8)

With , , and , a family of eigenfunctions is

(9)

where is a Jack polynomial in the monic normalization and is a partition with at most parts. Their eigenvalues are

(10)

This connects the integrable system to the differential equation for Jack polynomials (Lapointe and Vinet 1996).