We consider in L2(R) an elliptic second-order differential operator Aε, ε>0, given by Aε=-ddxg(x/ε)ddx+ε-2p(x/ε), with periodic coefficients. For small ε, we study the behavior of the resolvent of Aε at a regular point close to the edge of a spectral gap. We obtain an approximation of this resolvent in the “energy” norm with an error O(ε). The approximation is described in terms of the spectral characteristics of the operator at the edge of the gap. Bibliography: 22 titles.