Homogenization of a multidimensional periodic elliptic operator in a neighbourhood of the edge of the internal gap

The homogenization procedure for a multidimensional periodic Schrödinger operator near the edge of an internal gap is discussed. Approximation for the resolvent in the small period limit, with respect to the operator norm in $L_2(\mathbb{R}^d)$, is obtained. This approximation contains oscillations, but in a simpler form than the resolvent of the initial operator.