Integral formulas for wave functions of quantum many-body problems and representations of gl(n)

TL;DR

This paper derives explicit integral formulas for eigenfunctions of quantum many-body systems, specifically for Jack's symmetric functions, using Lie algebra representations and intertwining operators.

Contribution

It introduces a novel approach to express eigenfunctions via trace formulas involving Lie algebra modules, connecting quantum integrable systems with representation theory.

Findings

Explicit integral formulas for Jack's symmetric functions

Representation of eigenfunctions through traces of intertwining operators

Connection established between quantum many-body problems and Lie algebra modules

Abstract

We derive explicit integral formulas for eigenfunctions of quantum integrals of the Calogero-Sutherland-Moser operator with trigonometric interaction potential. In particular, we derive explicit formulas for Jack's symmetric functions. To obtain such formulas, we use the representation of these eigenfunctions by means of traces of intertwining operators between certain modules over the Lie algebra $\frak gl_n$, and the realization of these modules on functions of many variables.