Spectral approach to homogenization of nonstationary Schrödinger-type equations

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• 作者：Tatiana Suslina ^{t.suslina@spbu.ru" class="auth_mail" title="E-mail the corresponding author suslina@list.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Periodic differential operators ; Nonstationary Schr&ouml ; dinger-type equations ; Homogenization ; Effective operator ; Operator error estimates
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1466-1523
• 全文大小：1119 K

文摘

In L₂(R^d;Cⁿ)$L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, we consider selfadjoint strongly elliptic second order differential operators A_ε${\mathcal{A}}_{\epsilon }$ with periodic coefficients depending on ng>xng>/ε$\mathbf{x}/\epsilon$. We study the behavior of the operator exp⁡(−iA_ετ)$\exp (-i{\mathcal{A}}_{\epsilon }\tau )$, τ∈R$\tau \in \mathbb{R}$, for small ε  . Approximations for this exponential in the (H^s→L₂)$(H^s\to L_2)$-operator norm are obtained. The method is based on the scaling transformation, the Floquet–Bloch theory, and the analytic perturbation theory. The results are applied to study the behavior of the solution ng>ung>_ε${\mathbf{u}}_{\epsilon }$ of the Cauchy problem for the Schrödinger-type equation i∂_τng>ung>_ε=A_εng>ung>_ε+ng>Fng>$i{\partial }_{\tau }{\mathbf{u}}_{\epsilon }={\mathcal{A}}_{\epsilon }{\mathbf{u}}_{\epsilon }+\mathbf{F}$. Applications to the nonstationary Schrödinger equation and the two-dimensional Pauli equation with singular rapidly oscillating potentials are given.