Chern-Simons-Schrodinger theory on a one-dimensional lattice

TL;DR

This paper introduces a gauge-invariant Chern-Simons-Schrodinger system on a one-dimensional lattice, establishing well-posedness and analyzing stationary states with respect to lattice parameters.

Contribution

It develops a novel lattice version of the Chern-Simons-Schrodinger theory, proving well-posedness and studying stationary states without a variational framework.

Findings

Proved local and global well-posedness in square-summable space.

Analyzed existence regions of stationary bound states.

Addressed challenges due to lack of variational formulation.

Abstract

We propose a gauge-invariant system of the Chern-Simons-Schrodinger type on a one-dimensional lattice. By using the spatial gauge condition, we prove local and global well-posedness of the initial-value problem in the space of square summable sequences for the scalar field. We also study the existence region of the stationary bound states, which depends on the lattice spacing and the nonlinearity power. A major difficulty in the existence problem is related to the lack of variational formulation of the stationary equations. Our approach is based on the implicit function theorem in the anti-continuum limit and the solvability constraint in the continuum limit.