  We consider an elliptic differential operator $A_\varepsilon = - \frac{d}{dx}
g(x/\varepsilon) \frac{d}{dx} + \varepsilon^{-2} V(x/\varepsilon)$,
$\varepsilon > 0$, with periodic coefficients acting in $L_2(\mathbb{R})$. For
the nonstationary Schr\"{o}dinger equation with the Hamiltonian $A_\varepsilon$
and for the hyperbolic equation with the operator $A_\varepsilon$, analogs of
homogenization problems, related to the edges of the spectral bands of the
operator $A_\varepsilon$, are studied (the so called high-frequency
homogenization). For the solutions of the Cauchy problems for these equations
with special initial data, approximations in $L_2(\mathbb{R})$-norm for small
$\varepsilon$ are obtained.
