论文标题
具有周期性系数的高阶Schrödinger型方程的均质化
Homogenization of the higher-order Schrödinger-type equations with periodic coefficients
论文作者
论文摘要
在$ l_2（{\ Mathbb r}^d; {\ Mathbb c}^n）$中，我们考虑一个矩阵强烈椭圆形的差异差异运算符$ {a} _ \ varepsilon $ of订单$ 2P $，$ p \ peqslant 2 $。操作员$ {a} _ \ varepsilon $由$ {a} _ \ varepsilon = b（\ MathBf {d}）^* G（\ MathBf {x}/\ varepsilon）定期，有限和正面确定的矩阵值函数，以及$ b（\ mathbf {d}）$是订单$ p $的同质差分运算符。 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）$（带有合适的$ s $）$ l_2（{\ Mathbb r}^d; {\ Mathbb c}^n）$。这里$ a^0 $是有效的操作员。获得夏目误差估计值。结果适用于schrödinger-type方程的cauchy问题的均质化$ i \partial_τ{\ Mathbf u} _ \ varepsilon = {a} _ \ varepsilon { u} _ \ varepsilon \ vert_ {τ= 0} = \ boldsymbolϕ $。
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 $i \partial_τ{\mathbf u}_\varepsilon = {A}_\varepsilon {\mathbf u}_\varepsilon + {\mathbf F}$, ${\mathbf u}_\varepsilon\vert_{τ=0} = \boldsymbolϕ$.
