Solving the quantum non-linear Schrodinger equation with delta-type impurity

TL;DR

This paper provides an exact solution to the nonlinear Schrödinger equation with a delta impurity, using the RT algebra to incorporate particle-impurity interactions at quantum and classical levels.

Contribution

It introduces the use of the reflection-transmission algebra for solving the quantum nonlinear Schrödinger equation with a point impurity, extending previous methods.

Findings

Exact quantum scattering matrix derived

RT algebra captures particle-impurity interactions

Solution applies to classical and quantum cases

Abstract

We establish the exact solution of the nonlinear Schrodinger equation with a delta-function impurity, representing a point-like defect which reflects and transmits. We solve the problem both at the classical and the second quantized levels. In the quantum case the Zamolodchikov-Faddeev algebra, familiar from the case without impurities, is substituted by the recently discovered reflection-transmission (RT) algebra, which captures both particle-particle and particle-impurity interactions. The off-shell quantum solution is expressed in terms of the generators of the RT algebra and the exact scattering matrix of the theory is derived.