Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields

Peter D. Hislop, Eric Soccorsi

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


We study purely magnetic Schrödinger operators in two dimensions (x, y) with magnetic fields b(x) that depend only on the x-coordinate. The magnetic field b(x) is assumed to be bounded, there are constants 0<b-<b+<∞ such that b-≤b(x)≤b+, and outside of a strip of small width -ε<x<ε, where 0<ε<b-1/2-, we have b(x)=b±x for ±x>ε. The case of a jump in the magnetic field at x=0 corresponding to ε=0 is also studied. We prove that the magnetic field creates an effective barrier near x=0 that causes edge currents to flow along it consistent with the classical interpretation. We prove lower bounds on edge currents carried by states with energy localized inside the energy bands of the Hamiltonian. We prove that these edge current-carrying states are well-localized in x to a region of size b--1/2, also consistent with the classical interpretation. We demonstrate that the edge currents are stable with respect to various magnetic and electric perturbations. These lower bounds on the edge current hold for all time. For a family of perturbations compactly supported in the y-direction, we prove that the time asymptotic current exists and satisfies the same lower bound.

Original languageEnglish
Pages (from-to)594-624
Number of pages31
JournalJournal of Mathematical Analysis and Applications
Issue number1
StatePublished - Feb 1 2015

Bibliographical note

Publisher Copyright:
© 2014 Elsevier Inc.


  • Asymptotic velocity
  • Magnetic barrier
  • Magnetic edge states
  • Magnetic field
  • Schrödinger operators

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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