Homogenization of a one-dimensional periodic elliptic operator at the edge of a spectral gap: operator estimates in the energy norm

In $L_2(\mathbb{R})$, we consider an elliptic second-order differential operator $A_{\varepsilon}$, $\varepsilon >0$, given by $A_{\varepsilon} = - \frac{d}{dx} g(x/\varepsilon) \frac{d}{dx} + \varepsilon^{-2} p({x}/\varepsilon)$, with periodic coefficients. For small $\varepsilon$, we study the behavior of the resolvent of $A_{\varepsilon}$ in a regular point close to the edge of a spectral gap. We obtain approximation of this resolvent in the “energy” norm with error $O(\varepsilon)$. Approximation is described in terms of the spectral characteristics of the operator at the edge of the gap.