A Simple Lattice Version of the Nonlinear Schrodinger Equation and its Deformation with Exact Quantum Solution

TL;DR

This paper introduces a simplified lattice quantum nonlinear Schrödinger model that lacks certain symmetries but allows exact solutions and generalizations, providing new insights into quantum integrable systems.

Contribution

It presents a new, simpler lattice quantum NLS model without typical symmetries, enabling exact solutions and a natural vector generalization, along with a novel deformation involving q-boson operators.

Findings

Exact quantum solution via algebraic Bethe ansatz

Simplified Lax operator and conserved quantities

New deformation with Tamm-Dancoff like q-boson operators

Abstract

A lattice version of quantum nonlinear Schrodinger (NLS) equation is considered, which has significantly simple form and fullfils most of the criteria desirable for such lattice variants of field models. Unlike most of the known lattice NLS, the present model belongs to a class which does not exhibit the usual symmetry properties. However this lack of symmetry itself seems to be responsible for the remarkable simplification of the relevant objects in the theory, such as the Lax operator, the Hamiltonian and other commuting conserved quantities as well as their spectrum. The model allows exact quantum solution through algebraic Bethe ansatz and also a straightforward and natural generalisation to the vector case, giving thus a new exact lattice version of the vector NLS model. A deformation representing a new quantum integrable system involving Tamm-Dancoff like $q$-boson operators is constructed.