[Author]
 Dorodnyi, M. A.

[Title]
 High-energy homogenization of a multidimensional nonstationary Schr\"{o}dinger equation

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

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

[Abstract]
 In $L_2(\mathbb{R}^d)$, we consider an elliptic differential operator
 $\mathcal{A}_\varepsilon = - \operatorname{div} g(\mathbf{x}/\varepsilon) \nabla + 
 \varepsilon^{-2} V(\mathbf{x}/\varepsilon)$, $ \varepsilon > 0$, with periodic 
 coefficients. For the nonstationary Schr\"{o}dinger equation with the Hamiltonian 
 $\mathcal{A}_\varepsilon$, analogs of homogenization problems related to an arbitrary 
 point of the dispersion relation of the operator $\mathcal{A}_1$ are studied 
 (the so called high-energy homogenization).
 For the solutions of the Cauchy problems for these equations with special initial data, 
 approximations in $L_2(\mathbb{R}^d)$-norm for small $\varepsilon$ are obtained.

[Comments]
 Russian, 25 pp.

[Contact e-mail]
 mdorodni@yandex.ru