论文标题
非平稳施罗丁类型方程的均质均质化估计值：结果的清晰度
Operator error estimates for homogenization of the nonstationary Schrödinger-type equations: sharpness of the results
论文作者
论文摘要
在$ l_2（\ Mathbb {r}^d; \ Mathbb {C}^n）$中，我们考虑一个自偏连接矩阵，强烈椭圆形的二阶差差差速器$ \ MATHCAL {a} _ \ varepsilon $带有定期系数，该系数取决于$ \ Mathbf {x}}}}}/vareps/c。我们发现指数$ e^{ - iτ\ Mathcal {a} _ \ varepsilon} $，$τ\ in \ mathbb {r} $，对于小$ \ varepsilon $，in（$ h^s \ to l_2 $） - to l_2 $） - to l_2 $） - to l_2 $） - 适用$ $ s $。讨论了关于$τ$的错误估计的清晰度。结果适用于研究解决方案的行为$ \ mathbf {u} _ \ varepsilon $schrödinger-type方程$ i \ i \partial_τ\ partbf { + \ Mathbf {f} $。
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 $e^{-i τ\mathcal{A}_\varepsilon}$, $τ\in \mathbb{R}$, for small $\varepsilon$ in the ($H^s \to L_2$)-operator norm with suitable $s$. The sharpness of the error estimates with respect to $τ$ is discussed. The results are applied to study the behavior of the solution $\mathbf{u}_\varepsilon$ of the Cauchy problem for the Schrödinger-type equation $i\partial_τ \mathbf{u}_\varepsilon = \mathcal{A}_\varepsilon \mathbf{u}_\varepsilon + \mathbf{F}$.
