Polynomial algebras and exact solutions of general quantum non-linear optical models I: Two-mode boson systems

TL;DR

This paper develops polynomial algebra deformations to find exact solutions for two-mode boson quantum models, including BEC systems, by linking eigenfunctions to polynomial roots solving Bethe ansatz equations.

Contribution

It introduces higher order polynomial deformations of Lie algebra $A_1$, constructs their representations, and applies them to solve 2-mode boson systems exactly.

Findings

Eigenfunctions are polynomials with roots solving Bethe ansatz equations

Eigenvalues are expressed in terms of polynomial roots

Spectral equivalence between BEC models and quasi-exactly solvable potentials

Abstract

We introduce higher order polynomial deformations of $A_1$ Lie algebra. We construct their unitary representations and the corresponding single-variable differential operator realizations. We then use the results to obtain exact (Bethe ansatz) solutions to a class of 2-mode boson systems, including the Boson-Einstein Condensate models as special cases. Up to an overall factor, the eigenfunctions of the 2-mode boson systems are given by polynomials whose roots are solutions of the associated Bethe ansatz equations. The corresponding eigenvalues are expressed in terms of these roots. We also establish the spectral equivalence between the BEC models and certain quasi-exactly solvable Sch\"ordinger potentials.