Spectral Properties of Grain Boundaries at Small Angles of Rotation

TL;DR

This paper investigates the spectral properties of a 2D model for small-angle grain boundaries in crystals, showing how spectral gaps fill with spectrum as the rotation angle approaches zero and providing bounds for related density of states.

Contribution

It introduces a model for small-angle grain boundaries and analyzes how spectral gaps evolve, offering new insights into the spectral behavior of rotated periodic potentials.

Findings

Spectral gaps fill with spectrum as rotation angle approaches zero.

Bounds are established for the integrated density of states measure.

The model captures key spectral features of small-angle defects.

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 $V \colon \R^2 \to \R$, 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 $M_\theta \in \R^{2 \times 2}$ is the usual matrix describing rotation of the coordinates in $\R^2$ by an angle $\theta$. As a main result, it is shown that spectral gaps of the periodic Schr\"odinger operator $H_0 = -\Delta + V$ 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.