[Author]
Dorodnyi, M. A.
[Title]
Homogenization of the periodic Schr\"{o}dinger-type equations with the lower
order terms
[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 $\mathcal{B}_\varepsilon$,
$0<\varepsilon \le 1$, with periodic coeffcients depending on
$\mathbf{x}/\varepsilon$. The principal part of this operator is given in a
factorized form, the operator involves lower order terms. We find approximations
of the exponential $e^{-is \mathcal{B}_\varepsilon}$, $s \in \mathbb{R}$, for
small $\varepsilon$ in the ($H^r \to L_2$)-operator norm with suitable $r$.
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 $i\partial_{s} \mathbf{u}_\varepsilon = \mathcal{B}_\varepsilon
\mathbf{u}_\varepsilon + \mathbf{F}$. Applications to the magnetic
Schr\"{o}dinger equation with a singular electric potential and the
two-dimensional Pauli equation with singular potentials are given.
[Comments]
LaTeX, Russian, 69 pp.
[Contact e-mail]
mdorodni@yandex.ru