Integration formulas and exact calculations in the Calogero-Sutherland model

TL;DR

This paper discusses integration formulas used in exact calculations within the Calogero-Sutherland model, including inner product computations and a generalized Selberg integral, advancing analytical methods in this integrable system.

Contribution

It introduces and analyzes specific integration formulas crucial for exact correlation calculations in the Calogero-Sutherland model, including a generalized Selberg integral.

Findings

Derived formulas for inner products in the model

Extended the Selberg integral to new cases

Enhanced analytical tools for Calogero-Sutherland calculations

Abstract

Some integration formulas which either occur or are implicit in Ha's recent exact calculation of some correlations in the Calogero-Sutherland model are discussed. These integration formulas include the calculation of the inner product $<0|\rho (0)|\kappa>$ between the density operator acting on an excited state and the ground state, and a generalization of the Selberg integral due to Dotsenko and Fateev.