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:R2→R, we let Vθ(x,y)=V(x,y) in the right half-plane {x≥0} and Vθ=V∘M−θ in the left half-plane {x<0}, where Mθ∈R2×2 is the usual matrix describing rotation of the coordinates in R2 by an angle θ. As a main result, it is shown that spectral gaps of the periodic Schrödinger operator H0=−Δ+V fill with spectrum of Rθ=−Δ+Vθ 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.