Modeling a quantum Hall system via elliptic equations

TL;DR

This paper develops a novel PDE model for quantum Hall systems by replacing the Schrödinger equation with a Dirac-based elliptic system, advancing understanding of nanoscale quantum phenomena.

Contribution

It introduces a new elliptic PDE model for quantum Hall systems, replacing the Schrödinger equation with Dirac operator dynamics, highlighting quantum information aspects.

Findings

Formulation of a system of three coupled elliptic equations

Demonstration of the model's potential for nanosystem analysis

Extension of previous nonlinear feedback models

Abstract

Quantum Hall systems are a suitable theme for a case study in the general area of nanotechnology. In particular, it is a good framework in which to search for universal principles relevant to nanosystem modeling, and nanosystem-specific signal processing. Recently, we have been able to construct a PDE model of a quantum Hall system, which consists of the Schr\"odinger equation supplemented with a special type nonlinear feedback loop. This result stems from a novel theoretical approach, which in particular brings to the fore the notion of quantum information. In this article we undertake to modify the original model by substituting the dynamics based on the Dirac operator. This leads to a model that consists of a system of three nonlinearly coupled first order elliptic equations in the plane.