Green's Function Formulation of Multiple Nonlinear Dirac $\delta$-Function Potential in One Dimension

TL;DR

This paper develops a Green's function approach to analyze scattering and bound states in a one-dimensional nonlinear Dirac delta potential system with complex couplings, extending understanding of nonlinear quantum and optical models.

Contribution

It introduces a novel Green's function formulation for multiple nonlinear Dirac delta potentials with complex couplings, including bound state analysis for specific nonlinearities.

Findings

Derived explicit Green's function for the nonlinear Dirac delta system

Calculated bound state energies and wave functions for positive real couplings

Provided a framework for analyzing nonlinear scattering in quantum and optical contexts

Abstract

In this work, we study the scattering problem of the general nonlinear finitely many Dirac delta potentials with complex coupling constants (or opacities in the context of optics) using the Green's function method and then find the bound state energies and the wave functions for the particular form of the nonlinearity in the case of positive real coupling constants.