  We study the asymptotic behaviour, as the small parameter $\varepsilon$ tends
to zero, of the resolvents of uniformly elliptic second-order differential
operators with locally periodic coefficients depending on the slow variable $x$
and the fast variable $x/\varepsilon$, with periodicity only in the fast
variable. We provide a construction for the leading terms in the operator
asymptotics of these resolvents in the sense of $L^2$-operator-norm convergence
with order $\varepsilon^2$ remainder estimates. We apply the modified method of
the first approximation with the usage of the shift.
