The Initial Value Problem, Scattering and Inverse Scattering, for Non-Linear Schr\"odinger Equations with a Potential and a Non-Local Non-Linearity

TL;DR

This paper analyzes non-linear Schrödinger equations with potentials and non-local nonlinearities, establishing well-posedness, constructing scattering operators, and providing a method to uniquely reconstruct potentials and physical parameters from scattering data.

Contribution

It introduces a comprehensive study of initial value problems for these equations, including existence, uniqueness, and regularity, and develops a novel inverse scattering method for potential reconstruction.

Findings

Established local and global well-posedness for a broad class of potentials.

Constructed the scattering operator for potentials vanishing at infinity.

Developed a method to uniquely reconstruct potentials and physical parameters from scattering data.

Abstract

We consider non-linear Schr\"odinger equations with a potential, and non-local non-linearities, that are models in mesoscopic physics, for example of a quantum capacitor, and that also are models of molecular structure. We study in detail the initial value problem for these equations. In particular, existence and uniqueness of local and global solutions, continuous dependence on the initial data and regularity. We allow for a large class of unbounded potentials. We have no restriction on the growth at infinity of the positive part of the potential. We also construct the scattering operator in the case of potentials that go to zero at infinity. Furthermore, we give a method for the unique reconstruction of the potential from the small amplitude limit of the scattering operator. In the case of the quantum capacitor, our method allows us to uniquely reconstruct all the physical parameters from the small amplitude limit of the scattering operator.