How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2213  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax
  Remote Access

Topologically Protected States in One-Dimensional Systems

About this Title

C. L. Fefferman, J. P. Lee-Thorp and M. I. Weinstein

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 247, Number 1173
ISBNs: 978-1-4704-2323-0 (print); 978-1-4704-3707-7 (online)
Published electronically: February 1, 2017
Keywords:Schrödinger equation, Dirac equation, Floquet-Bloch theory, topological protection, edge states, Hill’s equation, domain wall

View full volume PDF

View other years and numbers:

Table of Contents


  • Chapter 1. Introduction and Outline
  • Chapter 2. Floquet-Bloch and Fourier Analysis
  • Chapter 3. Dirac Points of 1D Periodic Structures
  • Chapter 4. Domain Wall Modulated Periodic Hamiltonian and Formal Derivation of Topologically Protected Bound States
  • Chapter 5. Main Theorem—Bifurcation of Topologically Protected States
  • Chapter 6. Proof of the Main Theorem
  • Appendix A. A Variant of Poisson Summation
  • Appendix B. 1D Dirac points and Floquet-Bloch Eigenfunctions
  • Appendix C. Dirac Points for Small Amplitude Potentials
  • Appendix D. Genericity of Dirac Points - 1D and 2D cases
  • Appendix E. Degeneracy Lifting at Quasi-momentum Zero
  • Appendix F. Gap Opening Due to Breaking of Inversion Symmetry
  • Appendix G. Bounds on Leading Order Terms in Multiple Scale Expansion
  • Appendix H. Derivation of Key Bounds and Limiting Relations in the Lyapunov-Schmidt Reduction


We study a class of periodic Schrödinger operators, which in distinguished cases can be proved to have linear band-crossings or âĂIJDirac pointsâĂİ. We then show that the introduction of an âĂIJedgeâĂİ, via adiabatic modulation of these periodic potentials by a domain wall, results in the bifurcation of spatially localized âĂIJedge statesâĂİ. These bound states are associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. Our model captures many aspects of the phenomenon of topologically protected edge states for two-dimensional bulk structures such as the honeycomb structure of graphene. The states we construct can be realized as highly robust TM-electromagnetic modes for a class of photonic waveguides with a phase-defect.

References [Enhancements On Off] (What's this?)