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

### Tatiana A. Suslina

St. Petersburg State University, Russian Federation

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## Abstract

In $L_{2}(R_{d};C_{n})$, we consider a matrix strongly elliptic differential operator $A_{ε}$ of order $2p$, $p⩾2$. The operator $A_{ε}$ is given by $A_{ε}=b(D)_{∗}g(εx )b(D)$, $ε>0$, where $g(x)$ is a periodic, bounded, and positive definite matrix-valued function, and $b(D)$ is a homogeneous differential operator of order $p$. We prove that, for fixed $τ∈R$ and $ε→0$, the operator exponential $e_{−iτA_{ε}}$ converges to $e_{−iτA_{0}}$ in the norm of operators acting from the Sobolev space $H_{s}(R_{d};C_{n})$ (with a suitable $s$) into $L_{2}(R_{d};C_{n})$. 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 $i∂_{τ}u_{ε}=A_{ε}u_{ε}+F$, $u_{ε}∣_{τ=0}=ϕ$.