[Paper Review] On the eigenstates of the elliptic Calogero-Moser model

Kouichi Takemura|ArXiv.org|Feb 14, 2000

Algebraic structures and combinatorial models13 references4 citations

TL;DR

This paper establishes the convergence of the perturbative expansion for eigenstates and eigenvalues of the elliptic Calogero-Moser Hamiltonian in the limit of small $ au$-parameter (i.e., the trigonometric limit), using the Bethe Ansatz method. It proves that for $N=2$ with positive integer coupling $l$, and $N=3$ with $l=1$, the square-integrable eigenstates and eigenvalues of the elliptic model converge to those of the Calogero-Sutherland model, justifying regular perturbation in $p = \exp(2\pi i\tau)$.

ABSTRACT

It is known that the trigonometric Calogero-Sutherland model is obtained by the trigonometric limit (τ o \sqrt{-1} \infty) of the elliptic Calogero-Moser model, where (1,τ) 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 (2π\sqrt{-1} τ) 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 (2π\sqrt{-1} τ). With some assumptions, we show analogous results for N-particle and l is positive integer cases.

Motivation & Objective

• To justify the regular perturbation expansion of the elliptic Calogero-Moser model in the trigonometric limit, where $\tau \to i\infty$.
• To establish the existence of square-integrable eigenstates and eigenvalues for the elliptic model that converge to those of the Calogero-Sutherland model.
• To extend the convergence result from the $N=2$, $l \in \mathbb{Z}_{>0}$ and $N=3$, $l=1$ cases to general $N$-particle systems under certain assumptions.
• To connect the Bethe Ansatz solution of the elliptic model with the Jack polynomial eigenstates of the trigonometric model via analytic continuation in $p = \exp(2\pi i\tau)$.

Proposed method

• Uses the Bethe Ansatz method to reduce the eigenvalue problem of the elliptic Hamiltonian to solving transcendental Bethe Ansatz equations.
• Applies the implicit function theorem to analyze the analytic behavior of solutions near $p = 0$, corresponding to the trigonometric limit.
• Constructs eigenstates as formal power series in $p = \exp(2\pi i\tau)$, proving their convergence for $N=2$, $l \in \mathbb{Z}_{>0}$ and $N=3$, $l=1$.
• Implements (anti-)symmetrization of Bethe Ansatz states to ensure square-integrability, distinguishing them from finite-dimensional spaces of doubly periodic functions.
• Relies on the Pieri formula for Jack polynomials to compute matrix elements of perturbation terms $V_i$ in the Hamiltonian expansion $H = H_0 + \sum p^i V_i$.
• Uses the known eigenstate structure of the Calogero-Sutherland model (Jack polynomials) as a reference for the perturbative limit.

Experimental results

Research questions

• RQ1 Does the perturbative expansion of eigenstates of the elliptic Calogero-Moser model converge to the known eigenstates of the Calogero-Sutherland model in the trigonometric limit?
• RQ2 Under what conditions do the eigenstates of the elliptic model remain square-integrable and converge as $p = \exp(2\pi i\tau) \to 0$?
• RQ3 How does the solution of the Bethe Ansatz equation behave analytically near $p = 0$ for $N=2$ and $N=3$ with $l=1$?
• RQ4 Can the Bethe Ansatz method be used to construct eigenstates of the elliptic model that are not contained in the finite-dimensional space of doubly periodic functions?

Key findings

• For $N=2$ and $l \in \mathbb{Z}_{>0}$, all square-integrable eigenstates and eigenvalues of the elliptic Calogero-Moser model converge to those of the Calogero-Sutherland model as $p \to 0$.
• For $N=3$ and $l=1$, the convergence of eigenstates and eigenvalues is established, with explicit verification via analysis of the Bethe Ansatz equation and critical point behavior.
• The perturbation series in $p = \exp(2\pi i\tau)$ converges regularly for the $N=2$, $l \in \mathbb{Z}_{>0}$ and $N=3$, $l=1$ cases, justifying the use of formal perturbation methods.
• The eigenstates constructed via Bethe Ansatz are not in the finite-dimensional space of doubly periodic functions, and their (anti-)symmetrization ensures square-integrability.
• The method confirms that the diagonalization of the elliptic model's Hamiltonian is not directly linked to Jack polynomials with physical parameters, but rather to non-physical ones in the perturbative regime.
• The convergence result is extended to general $N$-particle systems under certain assumptions, particularly when $l$ is a positive integer.