Homogenization of nonstationary Schr\\"odinger type equations with periodic coefficients
T. A. Suslina · arXiv (Cornell University) · 2015
In $L_2(\\mathbb{R}^d;{\\mathbb C}^n)$ we consider selfadjoint strongly elliptic second order differential operators ${\\mathcal A}_\\varepsilon$ with periodic coefficients depending on ${\\mathbf x}/\\varepsilon$. We study the behavior of the operator exponential $\\exp(-i {\\mathcal A}_\\varepsilon \\tau)$, $\\tau \\in {\\mathbb R}$, for small $\\varepsilon$. Approximations for this exponential in the $(H^s\\to L_2)$-operator norm with a suitable $s$ are obtained. The results are applied to study the behavior of the solution ${\\mathbf u}_\\varepsilon$ of the Cauchy problem for the Schr\\"odinger type equation $i \\partial_\\tau {\\mathbf u}_\\varepsilon = {\\mathcal A}_\\varepsilon {\\mathbf u}_\\varepsilon$.