Edge States for generalised Iwatsuka models: Magnetic fields having a fast transition across a curve

TL;DR

This paper investigates the behavior of edge states in magnetic Laplacians with rapidly changing magnetic fields across a curve, extending previous models and providing detailed asymptotic analysis of eigenfunction localization.

Contribution

It generalizes Iwatsuka models to arbitrary regular curves and analyzes magnetic fields with slow variations, offering explicit descriptions of edge state mass distribution.

Findings

Extended edge state analysis to general curves

Derived explicit asymptotic mass distribution formulas

Included slowly varying magnetic field cases

Abstract

In this paper, we study the localization and propagation properties of the edge states associated to a class of magnetic laplacians in $\mathbb{R}^2$. We assume that the intensity of the magnetic field has a fast transition along a regular and compact curve $\Gamma$. Our main results extend to a general regular curve the study of the localised eigenfunction obtained when $\Gamma$ is a straight line (i.e. Iwatsuka models). Furthermore, we include in our analysis the case of magnetic fields that slowly change along the curve $\Gamma$ and we obtain a rigorous and explicit characterization of the asymptotic mass distribution of the edge state along $\Gamma$.