Spectral properties of grain boundaries at small angles of rotation

Abstract

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 W R 2 ! R, we let V # .x; y/ D V .x; y/ in the right half-plane fx 0g and V # D V B M # in the left half-plane fx < 0g, where M # 2 R 2 2 is the usual matrix describing rotation of the coordinates in R 2 by an angle #. As a main result, it is shown that spectral gaps of the periodic Schrödinger operator H D C V fill with spectrum of R # D C V # as 0 ¤ # ! 0. Moreover, we obtain upper and lower bounds for a quantity pertaining to an integrated density of states measure for the surface states. Mathematics Subject Classification (2010). Primary 35J10; Secondary 35P20, 81Q10.