Spectral Properties of Grain Boundaries at Small Angles of Rotation

We study some spectral properties of a simple two-dimensional model for small angle defects in crystals and alloys. Starting from a periodic potential $V \colon \R^2 \to \R$,welet$V_\theta(x,y) = V(x,y)$intherighthalf-plane${x \ge 0}$and$V_\theta = V \circ M_{-\theta}$inthelefthalf-plane${x < 0}$,where$M_\theta \in \R^{2 \times 2}$ 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 Schr"odinger operator $H_0 = -\Delta + V$fillwithspectrumof$R_\theta = -\Delta + V_\theta$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.