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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 $V \colon \R^2 \to \R$, we let $V_θ(x,y) = V(x,y)$ in the right half-plane $\{x \ge 0\}$ and $V_θ= V \circ M_{-θ}$ in the left half-plane $\{x < 0\}$, where $M_θ\in \R^{2 \times 2}$ 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 $0 \ne θ\to 0$. Moreover, we obtain upper and lower bounds for a quantity pertaining to an integrated density of states measure for the surface states.