**Memoirs of the American Mathematical Society**

2017;
118 pp;
Softcover

**Print ISBN: 978-1-4704-2323-0
Product Code: MEMO/247/1173**

List Price: $75.00

AMS Member Price: $45.00

MAA Member Price: $67.50

**Electronic ISBN: 978-1-4704-3707-7
Product Code: MEMO/247/1173.E**

List Price: $75.00

AMS Member Price: $45.00

MAA Member Price: $67.50

# Topologically Protected States in One-Dimensional Systems

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*C. L. Fefferman; J. P. Lee-Thorp; M. I. Weinstein*

The authors study a class of periodic Schrödinger operators, which in distinguished cases can be proved to have linear band-crossings or “Dirac points”. They then show that the introduction of an “edge”, via adiabatic modulation of these periodic potentials by a domain wall, results in the bifurcation of spatially localized “edge states”. These bound states are associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. The authors' 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 the authors construct can be realized as highly robust TM-electromagnetic modes for a class of photonic waveguides with a phase-defect.

#### Table of Contents

# Table of Contents

## Topologically Protected States in One-Dimensional Systems

- Cover Cover11
- Title page i2
- Chapter 1. Introduction and Outline 110
- Chapter 2. Floquet-Bloch and Fourier Analysis 1322
- Chapter 3. Dirac Points of 1D Periodic Structures 1726
- 3.1. The family of Hamiltonians, 𝐻(𝑠), and its Dirac points for 𝑠=1/2 1827
- 3.2. 𝐻(1/2)=-\Dₓ²+𝒬(𝓍;1/2) has an additional translation symmetry 1928
- 3.3. The action of -\Dₓ²+𝑉_{\ee}(𝑥) on 𝐿²_{𝑘_{⋆}=𝜋} 2029
- 3.4. Spectral properties of 𝐻^{(\eps=0)}=-\Dₓ² in 𝐿²_{𝑘} 2029
- 3.5. Sufficient conditions for occurrence of a 1D Dirac point 2130
- 3.6. Expansion of Floquet-Bloch eigenfunctions near a Dirac point 2231
- 3.7. Genericity of Dirac points at 𝑘=𝑘_{⋆} 2231

- Chapter 4. Domain Wall Modulated Periodic Hamiltonian and Formal Derivation of Topologically Protected Bound States 2534
- Chapter 5. Main Theorem—Bifurcation of Topologically Protected States 3342
- Chapter 6. Proof of the Main Theorem 3746
- 6.1. Rough strategy 3746
- 6.2. Detailed strategy: Decomposition into near and far energy components 3847
- 6.3. Analysis of far energy components 4049
- 6.4. Lyapunov-Schmidt reduction to a Dirac system for the near energy components 4453
- 6.5. Analysis of the band-limited Dirac system 5463
- 6.6. Proof of Proposition 6.10 5564
- 6.7. Final reduction to an equation for 𝜇=𝜇(𝛿) and its solution 6574

- Appendix A. A Variant of Poisson Summation 6978
- Appendix B. 1D Dirac points and Floquet-Bloch Eigenfunctions 7786
- Appendix C. Dirac Points for Small Amplitude Potentials 8190
- Appendix D. Genericity of Dirac Points - 1D and 2D cases 8594
- Appendix E. Degeneracy Lifting at Quasi-momentum Zero 93102
- Appendix F. Gap Opening Due to Breaking of Inversion Symmetry 97106
- Appendix G. Bounds on Leading Order Terms in Multiple Scale Expansion 101110
- Appendix H. Derivation of Key Bounds and Limiting Relations in the Lyapunov-Schmidt Reduction 105114
- References 117126
- Back Cover Back Cover1132