# Spectral properties of grain boundaries at small angles of rotation

### Rainer Hempel

Technische Universität Braunschweig, Germany### Martin Kohlmann

Leibniz Universität Hannover, Germany

## Abstract

We study some spectral properties of a simple two-dimensional model for small angle defects in crystals and alloys. Starting from a periodic potential $V:R_{2}→R$, we let $V_{θ}(x,y)=V(x,y)$ in the right half-plane ${x≥0}$ and $V_{θ}=V∘M_{−θ}$ in the left half-plane ${x<0}$, where $M_{θ}∈R_{2×2}$ is the usual matrix describing rotation of the coordinates in $R_{2}$ by an angle $θ$. As a main result, it is shown that spectral gaps of the periodic Schrödinger operator $H_{0}=−Δ+V$ fill with spectrum of $R_{θ}=−Δ+V_{θ}$ as $0=θ→0$. Moreover, we obtain upper and lower bounds for a quantity pertaining to an integrated density of states measure for the surface states.

## Cite this article

Rainer Hempel, Martin Kohlmann, Spectral properties of grain boundaries at small angles of rotation. J. Spectr. Theory 1 (2011), no. 2, pp. 197–219

DOI 10.4171/JST/9