Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A method to derive explicit formulas for an elliptic generalization of the Jack polynomials

Edwin Langmann

TL;DR

The paper develops an explicit procedure to obtain closed-form expressions for Jack polynomials by linking them to the eigenfunctions of the Sutherland model, casting the problem into a Fourier-type transform framework with an explicit contour-integral construction. It further outlines a generalization to the elliptic Calogero-Sutherland (eCS) model, where symmetric functions generalize Jack polynomials via elliptic theta functions and an implicit eigenvalue equation solved through a perturbative Lagrange-inversion approach in the elliptic parameter $q$. The main contributions include an explicit eigenfunction structure for the Sutherland model, a finite-contour integral representation for the symmetric functions $f_{\mathbf{m}}$, and a scalable path to explicit, series-based eigenvalues and eigenfunctions in the elliptic case. This work provides a bridge between integrable quantum many-body systems and explicit symmetric-polynomial formulas, with potential connections to Macdonald polynomials and Ruijsenaars models for broader applicability.

Abstract

We review a method providing explicit formulas for the Jack polynomials. Our method is based on the relation of the Jack polynomials to the eigenfunctions of a well-known exactly solvable quantum many-body system of Calogero-Sutherland type. We also sketch a generalization of our method allowing to find the exact solution of the elliptic generalization of the Calogero-Sutherland model. We present the resulting explicit formulas for certain symmetric functions generalizing the Jack polynomials to the elliptic case.

A method to derive explicit formulas for an elliptic generalization of the Jack polynomials

TL;DR

The paper develops an explicit procedure to obtain closed-form expressions for Jack polynomials by linking them to the eigenfunctions of the Sutherland model, casting the problem into a Fourier-type transform framework with an explicit contour-integral construction. It further outlines a generalization to the elliptic Calogero-Sutherland (eCS) model, where symmetric functions generalize Jack polynomials via elliptic theta functions and an implicit eigenvalue equation solved through a perturbative Lagrange-inversion approach in the elliptic parameter . The main contributions include an explicit eigenfunction structure for the Sutherland model, a finite-contour integral representation for the symmetric functions , and a scalable path to explicit, series-based eigenvalues and eigenfunctions in the elliptic case. This work provides a bridge between integrable quantum many-body systems and explicit symmetric-polynomial formulas, with potential connections to Macdonald polynomials and Ruijsenaars models for broader applicability.

Abstract

We review a method providing explicit formulas for the Jack polynomials. Our method is based on the relation of the Jack polynomials to the eigenfunctions of a well-known exactly solvable quantum many-body system of Calogero-Sutherland type. We also sketch a generalization of our method allowing to find the exact solution of the elliptic generalization of the Calogero-Sutherland model. We present the resulting explicit formulas for certain symmetric functions generalizing the Jack polynomials to the elliptic case.

Paper Structure

This paper contains 4 sections, 6 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Solution of the Sutherland model
  3. Solution of the elliptic Calogero-Sutherland model: Outline
  4. Final remarks

Key Result

Theorem 2.1

For ${\bf m}\in{\mathbb Z}^N$, let where and the integration contours ${\mathcal{C}}_j$ are nested circles in the complex plane enclosing the unit circle, for some $\varepsilon>0$. Moreover, for partitions ${\bf n}$, let with where for all ${\bf m}\in{\mathbb Z}^N$, ${\bf E}_{jk}$ is the vector in ${\mathbb Z}^N$ with the following components, for $j,k,\ell=1,2,\ldots,N$, and $\delta({\bf n

Theorems & Definitions (9)

  • Theorem 2.1
  • Lemma 2.2
  • proof : Proof of Theorem \ref{['thm1']}
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof : Proof of Lemma \ref{['lemma0']}
  • Theorem 3.1
  • Lemma 4.1