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Edge states for second order elliptic operators in a channel

David Gontier

Abstract

We present a general framework to study edge states for second order elliptic operators in a half channel. We associate an integer valued index to some bulk materials, and we prove that for any junction between two such materials, localised states must appear at the boundary whenever the indices differ.

Edge states for second order elliptic operators in a channel

Abstract

We present a general framework to study edge states for second order elliptic operators in a half channel. We associate an integer valued index to some bulk materials, and we prove that for any junction between two such materials, localised states must appear at the boundary whenever the indices differ.

Paper Structure

This paper contains 38 sections, 32 theorems, 235 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. First facts and notations
  3. Lagrangian planes in complex Hilbert spaces
  4. Basics in symplectic spaces
  5. Lagrangian planes of Hilbert spaces and unitaries.
  6. Another unitary
  7. Self-adjoint extensions of Hill's operators
  8. Self-adjoint operators
  9. Self-adjoint extensions of Hill's operators on the semi line
  10. The Lagrangian planes $\ell^\pm(E)$
  11. Families of Hill's operators
  12. Families of Lagrangians, and Maslov index
  13. Definition with quadratic crossing forms
  14. Definition with the unitaries $U$
  15. Definition with the unitary ${\mathcal{U}}$.
  16. ...and 23 more sections

Key Result

Theorem 1

Let $t \mapsto V_{R,t}$ and $t \mapsto V_{L,t}$ be two continuous periodic families of bounded potentials on ${\mathbb R}$. Let $E \in {\mathbb R}$ be in the gap of both corresponding (bulk) Hill's operators $(h_{L, t})$ and $(h_{R, t})$. Let $\chi : {\mathbb R} \to [0, 1]$ be any switch function, s Then

Theorems & Definitions (68)

  • Theorem 1: Junctions between two channels
  • Example 2: In ${\mathbb R}^{2n}$
  • Example 3: In ${\mathbb C}^{2n}$
  • Example 4: In ${\mathbb C}^N$
  • Lemma 5
  • proof
  • Example 6
  • Lemma 7
  • Corollary 8
  • Lemma 9
  • ...and 58 more