Spectral properties of grain boundaries at small angles of rotation

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 $V:{\mathbb{R}}^2\to \mathbb{R}$, we let $V_{\theta }(x,y)=V(x,y)$ in the right half-plane $\{x\ge 0\}$ and $V_{\theta }=V\circ M_{-\theta }$ in the left half-plane $\{x<0\}$, where $M_{\theta }\in {\mathbb{R}}^{2\times 2}$ is the usual matrix describing rotation of the coordinates in ${\mathbb{R}}^2$ by an angle $\theta$. As a main result, it is shown that spectral gaps of the periodic Schrödinger operator $H_0=-\Delta +V$ fill with spectrum of $R_{\theta }=-\Delta +V_{\theta }$ as $0\ne \theta \to 0$. Moreover, we obtain upper and lower bounds for a quantity pertaining to an integrated density of states measure for the surface states.