On the eigenstates of the elliptic Calogero-Moser model

TL;DR

This paper demonstrates that for certain eigenstates of the Calogero-Sutherland model, there exist corresponding eigenstates of the elliptic Calogero-Moser model that converge to them as the elliptic parameter approaches a limit, justifying a perturbation approach.

Contribution

The paper establishes the existence and convergence of elliptic Calogero-Moser eigenstates to Calogero-Sutherland eigenstates for specific particle and coupling cases, validating perturbation methods.

Findings

Eigenstates of the elliptic model converge to those of the trigonometric model under certain conditions.

Justification of regular perturbation with respect to the elliptic parameter.

Results extended to N-particle and integer coupling cases under assumptions.

Abstract

It is known that the trigonometric Calogero-Sutherland model is obtained by the trigonometric limit (\tau \to \sqrt{-1} \infty) of the elliptic Calogero-Moser model, where (1,\tau) is a basic period of the elliptic function. We show that for all square-integrable eigenstates and eigenvalues of the Hamiltonian of the Calogero-Sutherland model, if \exp (2\pi \sqrt{-1} \tau ) is small enough then there exist square-integrable eigenstates and eigenvalues of the Hamiltonian of the elliptic Calogero-Moser model which converge to the ones of the Calogero-Sutherland model for the 2-particle and the coupling constant l is positive integer cases and the 3-particle and l=1 case. In other words, we justify the regular perturbation with respect to the parameter \exp (2\pi \sqrt{-1} \tau). With some assumptions, we show analogous results for N-particle and l is positive integer cases.