An indefinite system of equations governing the fractional quantum Hall effect

TL;DR

This paper proves the existence of solutions to a complex indefinite system of equations related to the fractional quantum Hall effect, extending solutions from bounded to full-plane domains with decay analysis.

Contribution

It introduces a novel approach using partial coercivity to establish solutions for indefinite coupled nonlinear systems in quantum physics.

Findings

Existence of solutions on bounded and full-plane domains

Asymptotic decay behavior of solutions

Conditions for solutions in doubly-periodic domains

Abstract

In this paper, we establish existence of solutions to an indefinite coupled non-linear system. We use partial coercivity to establish existence of a critical point to an indefinite functional and thus the existence of solutions on both bounded domains and doubly-periodic domains. We then take the limit of the bounded domain to extend solutions on a bounded domain to the full-plane. We also provide asymptotic decay analysis for solutions over the full-plane. Necessary and sufficient conditions are also provided for the doubly periodic domain.