Dislocation problems for periodic Schr\"odinger operators and mathematical aspects of small angle grain boundaries

TL;DR

This paper investigates spectral effects of dislocations and small angle defects in two-dimensional periodic Schrödinger operators, revealing how such defects introduce spectrum into gaps and fill spectral gaps as defect angles diminish.

Contribution

It provides new insights into how dislocations and small angle grain boundaries affect the spectrum of periodic Schrödinger operators, including estimates for surface states and spectral filling.

Findings

Dislocations produce spectrum inside the gaps of the periodic operator.

Small angle defects cause the spectral gaps to fill with spectrum as the angle approaches zero.

Estimates for the density of surface states associated with dislocations.

Abstract

We discuss two types of defects in two-dimensional lattices, namely (1) translational dislocations and (2) defects produced by a rotation of the lattice in a half-space. For Lipschitz-continuous and $\Z^2$-periodic potentials, we first show that translational dislocations produce spectrum inside the gaps of the periodic problem; we also give estimates for the (integrated) density of the associated surface states. We then study lattices with a small angle defect where we find that the gaps of the periodic problem fill with spectrum as the defect angle goes to zero. To introduce our methods, we begin with the study of dislocation problems on the real line and on an infinite strip. Finally, we consider examples of muffin tin type. Our overview refers to results in [HK1, HK2].