Normalization of the wavwfunction for the Calogero-Sutherland model with internal degrees of freedom

TL;DR

This paper conjectures the exact normalization of a multicomponent Calogero-Sutherland wavefunction, extending Selberg integrals and providing new proofs for these integrals in special cases.

Contribution

It introduces a conjecture for the normalization of multicomponent wavefunctions and offers a new proof of the Selberg integral, extending it to a two-component case.

Findings

Conjectured normalization formula for multicomponent wavefunctions.

New proof of the Selberg integral.

Extension of Selberg integral to two-component case.

Abstract

The exact normalization of a multicomponent generalization of the ground state wavefunction of the Calogero-Sutherland model is conjectured. This result is obtained from a conjectured generalization of Selberg's $N$-dimensional extension of the Euler beta integral, written as a trigonometric integral. A new proof of the Selberg integral is given, and the method is used to provide a proof of the mulicomponent generalization in a special two-component case.