Analytic solutions of the 1D finite coupling delta function Bose gas

TL;DR

This paper provides exact analytic solutions for the 1D finite coupling delta function Bose gas, detailing ground state properties across coupling regimes using Bethe ansatz, and offers new insights into its correlation functions and structure factors.

Contribution

It presents new analytic solutions for the 1D Bose gas at finite coupling, including detailed results for specific particle numbers and general N, expanding understanding of its ground state properties.

Findings

Weak coupling density matrix has a rational polynomial structure.

Explicit results for finite N and general N are derived.

Key quantities like occupation numbers and structure factor are characterized.

Abstract

An intensive study for both the weak coupling and strong coupling limits of the ground state properties of this classic system is presented. Detailed results for specific values of finite $N$ are given and from them results for general $N$ are determined. We focus on the density matrix and concomitantly its Fourier transform, the occupation numbers, along with the pair correlation function and concomitantly its Fourier transform, the structure factor. These are the signature quantities of the Bose gas. One specific result is that for weak coupling a rational polynomial structure holds despite the transcendental nature of the Bethe equations. All these new results are predicated on the Bethe ansatz and are built upon the seminal works of the past.