Homogenization of a multidimensional periodic elliptic operator at the edge of a spectral gap: Operator estimates in the energy norm

We consider an elliptic differential operator $\mathcal{A}_{\varepsilon} = - \operatorname{div}\widetilde{g}(\boldsymbol{x}/\varepsilon) \nabla+ \varepsilon^{-2} V(\boldsymbol{x}/\varepsilon),$ $ \varepsilon > 0$, with periodic coefficients acting in $L_2(\mathbb{R}^{d})$. The spectrum of a periodic operator $\mathcal{A}_{\varepsilon}$ has a band structure and may have gaps. For small $ \varepsilon$, we study the behavior of the resolvent of the operator $ \mathcal{A}_{\varepsilon} $ in a regular point close to the edge of a spectral gap. We obtain an approximation of this resolvent in the “energy” norm with error $ O(\varepsilon)$.