High-frequency homogenization of nonstationary periodic equations

Abstract: 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.