Operator estimates for homogenization of the two-dimensional Dirac operator with periodic coefficients

The paper concerns the homogenization theory of periodic differential operators. We consider a two-dimensional self-adjoint Dirac operator $\mathcal{D}_{\epsilon}$ with a singular magnetic potential $\epsilon^{-1}\mathbf{A}(x/\epsilon)$ and a matrix potential $V(x/\epsilon)$, where $\epsilon>0$ is a small parameter. The coefficients $\mathbf{A}$ and $V$ are assumed to be periodic. The behavior of the resolvent $(\mathcal{D}_{\epsilon}-i I)^{-1}$ for small $\epsilon$ is studied. We obtain an approximation for the resolvent in the operator norm on $L_2(\mathbb{R}^2)$ with a sharp order error estimate. The approximation is given by the resolvent of the effective operator with constant coefficients sandwiched between rapidly oscillating factors.