The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions

TL;DR

This paper rigorously derives the one-dimensional Lieb-Liniger model from a three-dimensional dilute Bose gas by analyzing eigenvalues and eigenfunctions in elongated traps, establishing a precise limit for the interaction parameters.

Contribution

It provides a rigorous mathematical derivation of the Lieb-Liniger model as a limit of 3D dilute bosons with finite scattering length, including uniform bounds on eigenvalues and eigenfunctions.

Findings

Lieb-Liniger model obtained as a limit of 3D Bose gas

Uniform bounds on eigenvalues and eigenfunctions

Derivation valid for all coupling constants g in [0, ∞]

Abstract

We show that the Lieb-Liniger model for one-dimensional bosons with repulsive $\delta$-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length $a$ and the radius $r$ of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant $g \sim a/r^2$ is derived. Our bounds are uniform in $g$ in the whole parameter range $0\leq g\leq \infty$, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size $\sim r^{-2}$ above the ground state energy.