Gauge momentum operators for the Calogero-Sutherland model with anti-periodic boundary condition

TL;DR

This paper investigates the integrability of the anti-periodic Calogero system, constructs gauge momentum operators, and proposes a general scheme applicable to various boundary conditions, confirming the system's integrability.

Contribution

It introduces a method to construct hermitian gauge momentum operators for the anti-periodic Calogero-Sutherland model, extending to both periodic and anti-periodic boundary conditions.

Findings

Gauge momentum operators are hermitian and diagonalizable with the Hamiltonian.

The scheme applies to trigonometric and hyperbolic Calogero-Sutherland models.

Existence of these operators confirms the system's integrability.

Abstract

The integrability of a classical Calogero systems with anti-periodic boundary condition is studied. This system is equivalent to the periodic model in the presence of a magnetic field. Gauge momentum operators for the anti-periodic Calogero system are constructed. These operators are hermitian and simultaneously diagonalizable with the Hamiltonian. A general scheme for constructing such momentum operators for trigonometric and hyperbolic Calogero-Sutherland model is proposed. The scheme is applicable for both periodic and anti-periodic boundary conditions. The existence of these momentum operators ensures the integrability of the system. The interaction parameter $\lambda$ is restricted to a certain subset of real numbers. This restriction is in fact essential for the construction of the hermitian gauge momentum operators.