Delta-Bose Gas: the Matrix Elements of the Particle Number Operator as a Determinant

TL;DR

This paper reviews the Coordinate Bethe Ansatz for the delta-Bose gas and derives a determinant expression for the matrix elements of the particle number operator, confirming a conjecture by V. Terras.

Contribution

It provides a detailed review and a new determinant formula for matrix elements of the particle number operator in the delta-Bose gas, advancing analytical methods in integrable models.

Findings

Derived a determinant expression for matrix elements

Confirmed V. Terras's conjecture

Enhanced understanding of Bethe Ansatz applications

Abstract

We first propose a pedestrian review of the Coordinate Bethe Ansatz for the $\delta$-Bose gas on an infinite axis. The scalar product of Bethe states, briefly reviewed, provides the first keystone to compute a compact expression for the Matrix Elements of the Particle Number Operator as conjectured by V. Terras, the main result of the following reflection.