The Nonlinear Schrodinger Equation on the Half Line

TL;DR

This paper investigates the nonlinear Schrödinger equation on a half line with mixed boundary conditions, introducing a new algebraic structure called boundary algebra to construct an exact quantum solution and analyze scattering properties.

Contribution

It introduces boundary algebra as a novel algebraic framework for solving the nonlinear Schrödinger equation with boundaries, extending quantum field theory methods.

Findings

Constructed an exact second quantized solution using boundary algebra.

Established quantum properties of the solution and derived the scattering operator.

Generalized Haag-Ruelle framework for broken translation invariance.

Abstract

The nonlinear Schrodinger equation on the half line with mixed boundary condition is investigated. After a brief introduction to the corresponding classical boundary value problem, the exact second quantized solution of the system is constructed. The construction is based on a new algebraic structure, which is called in what follows boundary algebra and which substitutes, in the presence of boundaries, the familiar Zamolodchikov-Faddeev algebra. The fundamental quantum field theory properties of the solution are established and discussed in detail. The relative scattering operator is derived in the Haag-Ruelle framework, suitably generalized to the case of broken translation invariance in space.