Asymmetric Bethe Ansatz

TL;DR

This paper introduces the Asymmetric Bethe Ansatz, a novel method that relaxes traditional integrability conditions, enabling exact solutions for certain non-integrable quantum systems, exemplified by a bosonic dimer in a delta-well.

Contribution

The paper proposes the Asymmetric Bethe Ansatz, allowing exact solutions for systems that violate conventional Bethe Ansatz integrability conditions.

Findings

The Asymmetric Bethe Ansatz generalizes traditional methods.

Exact solutions are found for a bosonic dimer in a delta-well.

The Liu-Qi-Zhang-Chen problem is a special case of this new approach.

Abstract

The recently proposed exact quantum solution for two $\delta$-function-interacting particles with a mass-ratio $3\!:\!1$ in a hard-wall box [Y. Liu, F. Qi, Y. Zhang and S. Chen, iScience 22, 181 (2019)] violates the conventional necessary condition for a Bethe Ansatz integrability, the condition being that the system must be reducible to a superposition of semi-transparent mirrors that is invariant under all the reflections it generates. In this article, we found a way to relax this condition: some of the semi-transparent mirrors of a known self-invariant mirror superposition can be replaced by the perfectly reflecting ones, thus breaking the self-invariance. The proposed name for the method is \emph{Asymmetric Bethe Ansatz} (Asymmetric BA). As a worked example, we study in detail the bound states of the nominally non-integrable system comprised of a bosonic dimer in a $\delta$-well. Finally, we show that the exact solution of the Liu-Qi-Zhang-Chen problem is a particular instance of the the Asymmetric BA.