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arXiv 2011.13382math.AP
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周期系数高阶Schrödinger型方程的均匀化

Homogenization of the higher-order Schrödinger-type equations with periodic coefficients

Tatiana Suslina

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22 pages
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摘要：

在$L_2({\mathbb R}^d; {\mathbb C}^n)$中，我们考虑了阶为$2p$，$p \geqslant 2$的矩阵强椭圆型微分算子${A}_\varepsilon$.算子${A}_\varepsilon$由${A}_\varepsilon = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D})$，$\varepsilon >0$给出，其中$g(\mathbf{x})$是周期的、有界的、正定的矩阵值函数，$b(\mathbf{D})$是阶为$p$的齐次微分算子.我们证明了，对于固定的$τ\in {\mathbb R}$和$\varepsilon \to 0$，算子指数$e^{-i τ{A}_\varepsilon}$收敛到$e^{-i τ{A}^0}$的运营商从Sobolev空间$H^s({\mathbb R}^d; {\mathbb C}^n)$（与一个合适的$s$）到$L_2({\mathbb R}^d; {\mathbb C}^n)$的范数。这里$A^0$是有效的操作符。得到了锐阶误差估计。结果被应用到均匀化的柯西问题的薛定谔型方程$i \partial_τ{\mathbf u}_\varepsilon = {A}_\varepsilon {\mathbf u}_\varepsilon + {\mathbf F}$，${\mathbf u}_\varepsilon\vert_{τ=0} = \boldsymbolϕ$。

英文摘要：

In $L_2({\mathbb R}^d; {\mathbb C}^n)$, we consider a matrix strongly elliptic differential operator ${A}_\varepsilon$ of order $2p$, $p \geqslant 2$. The operator ${A}_\varepsilon$ is given by ${A}_\varepsilon = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D})$, $\varepsilon >0$, where $g(\mathbf{x})$ is a periodic, bounded, and positive definite matrix-valued function, and $b(\mathbf{D})$ is a homogeneous differential operator of order $p$. We prove that, for fixed $τ\in {\mathbb R}$ and $\varepsilon \to 0$, the operator exponential $e^{-i τ{A}_\varepsilon}$ converges to $e^{-i τ{A}^0}$ in the norm of operators acting from the Sobolev space $H^s({\mathbb R}^d; {\mathbb C}^n)$ (with a suitable $s$) into $L_2({\mathbb R}^d; {\mathbb C}^n)$. Here $A^0$ is the effective operator. Sharp-order error estimate is obtained. The results are applied to homogenization of the Cauchy problem for the Schrödinger-type equation $i \partial_τ{\mathbf u}_\varepsilon = {A}_\varepsilon {\mathbf u}_\varepsilon + {\mathbf F}$, ${\mathbf u}_\varepsilon\vert_{τ=0} = \boldsymbolϕ$.

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