Resolvent approximations of periodic elliptic operators of even order $2m\ge 6$

For divergence form selfadjoint elliptic operators with $\varepsilon$-periodic coefficients of arbitrary even order $2m\ge 6$, we construct resolvent approximations in the operator energy norm $\|\cdot\|_{L^2{\to}H^m}$ with an error of the order $\varepsilon^3$ as $\varepsilon\to 0$. We consider scalar operators with real-valued coefficients and apply two-scale expansion method combined with smoothing technique.