Homogenization of the higher-order Schr\\"odinger-type equations with periodic coefficients
T. A. Suslina · arXiv (Cornell University) · 2020
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 $\\tau \\in {\\mathbb R}$ and $\\varepsilon \\to 0$, the operator exponential $e^{-i \\tau {A}_\\varepsilon}$ converges to $e^{-i \\tau {A}^0}$ 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 Schr\\"odinger-type equation $i \\partial_\\tau {\\mathbf u}_\\varepsilon = {A}_\\varepsilon {\\mathbf u}_\\varepsilon + {\\mathbf F}$, ${\\mathbf u}_\\varepsilon\\vert_{\\tau=0} = \\boldsymbol{\\phi}$.