[Author]
 Dorodnyi, M. A.

[Title]
 Operator error estimates for homogenization of the nonstationary
 Schr\"{o}dinger-type equations: dependence on time

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

[Keywords]
 Periodic differential operators, Nonstationary Schr\"{o}dinger-type equations,
 Homogenization, Effective operator, Operator error estimates

[Abstract]
 In $L_2(\mathbb{R}^d; \mathbb{C}^n)$, we consider a selfadjoint matrix strongly
 elliptic second order differential operator $\mathcal{A}_\varepsilon$, with 
 periodic coefficients depending on $\mathbf{x}/\varepsilon$. 
 We find approximations of the exponential $e^{-i \tau \mathcal{A}_\varepsilon}$, 
 $\tau \in \mathbb{R}$, for small $\varepsilon$ in the ($H^s \to L_2$)-operator 
 norm with suitable $s$. The sharpness of the error estimates with respect to 
 $\tau$ is discussed. The results are applied to study the behavior 
 of the solution $\mathbf{u}_\varepsilon$ of the Cauchy problem for the
 Schr\"{o}dinger-type equation $i\partial_{\tau} \mathbf{u}_\varepsilon = 
 mathcal{A}_\varepsilon \mathbf{u}_\varepsilon + \mathbf{F}$.
 
[Comments]
 LaTeX, Russian, 44 pp.

[Contact e-mail]
 mdorodni@yandex.ru