Perturbative Analysis of Potential Scattering Problems in the Lieb-Liniger Model

TL;DR

This paper performs a first-order perturbation analysis on the Lieb-Liniger model with an external potential, revealing the system's insensitivity due to many-body interactions and providing insights into eigenstate shifts and density expectations.

Contribution

It introduces a perturbative approach using Bethe states to analyze eigenenergy shifts and density expectations in the Lieb-Liniger model with external potential.

Findings

Eigenstates can resemble Schrödinger's cat states in large systems

Many-body interactions make the system insensitive to external potential

Density operator expectations can be computed using Slavnov's formula

Abstract

The Lieb-Liniger model which has a weak external potential term under the periodic boundary condition is investigated. By exploiting the Bethe states as bases, we perform a perturbation analysis up to the first order to obtain the shifts of eigenenergies and corresponding eigenstates which have been brought about by the external potential. If we take a sufficiently large system, the eigenstates can be "the Schr\"odinger's cat states". Expectation values of the density operator taken between two Bethe states can be calculated with the aid of the Slavnov's formula and we evaluate the influence of the many-body interaction to the system under the external potential. The system is insensitive to the external potential because of the many-body interaction.