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 $V \colon \Bbb R^2 \to \Bbb R$, 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 $M_\theta \in \Bbb R^{2 \times 2}$ is the usual matrix describing rotation of the coordinates in $\Bbb R^2$ 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.