[Author]
 Dorodnyi M. A.

[Title]
 High-frequency homogenization of nonstationary periodic equations

[AMS Subj-class]
 35B27 Homogenization; partial differential equations in media with periodic structure

[Keywords]
 periodic differential operators, Schr\"{o}dinger-type equations, 
 hyperbolic equations, spectral bands, homogenization, 
 effective operator, operator error estimates

[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  (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.

[Comments]
 LaTeX, Russian, 25 pp.

[Contact e-mail]
 mdorodni@yandex.ru