# Edge states in ordinary differential equations for dislocations

## TL;DR

This paper investigates the spectral properties of Schrödinger and Dirac operators with dislocation potentials on the real line, establishing a bulk-edge correspondence through integer-valued indices and analyzing eigenvalues related to dislocations.

## Contribution

It introduces and proves the equivalence of various spectral indices, including Chern numbers and spectral flows, for dislocations in differential operators, demonstrating a bulk-edge correspondence.

## Key findings

- All introduced indices coincide, confirming bulk-edge correspondence.
- Zero eigenvalue always exists for dislocated Dirac operators.
- Spectral variations depend on the dislocation parameter.

## Abstract

In this article, we study Schr\"odinger operators on the real line, when the external potential represents a dislocation in a periodic medium. We study how the spectrum varies with the dislocation parameter. We introduce several integer-valued indices, including Chern number for bulk indices, and various spectral flows for edge indices. We prove that all these indices coincide, providing a proof a bulk-edge correspondence in this case. The study is also made for dislocations in Dirac models on the real line. We prove that 0 is always an eigenvalue of such operators.