In \(L_2({\mathbb {R}}^d;{\mathbb {C}}^n)\), we consider a semigroup \(e^{-tA_\varepsilon }\), \(t\geqslant 0\), generated by a matrix elliptic second-order differential operator \(A_\varepsilon \geqslant 0\). Coefficients of \(A_\varepsilon \) are periodic, depend on \({\mathbf {x}}/\varepsilon \), and oscillate rapidly as \(\varepsilon \rightarrow 0\). Approximations for \(e^{-tA_\varepsilon }\) were obtained by Suslina (Funktsional Analiz i ego Prilozhen 38(4):86–90, 2004) and Suslina (Math Model Nat Phenom 5(4):390–447, 2010) via the spectral method and by Zhikov and Pastukhova (Russ J Math Phys 13(2):224–237, 2006) via the shift method. In the present note, we give another short proof based on the contour integral representation for the semigroup and approximations for the resolvent with two-parametric error estimates obtained by Suslina (2015).

在\(L_2({\mathbb {R}}^d;{\mathbb {C}}^n)\) 中，我们考虑一个半群\(e^{-tA_\varepsilon }\)，\(t\geqslant 0\)，由矩阵椭圆二阶微分算子\(A_\varepsilon \geqslant 0\) 生成。\(A_\varepsilon \) 的系数是周期性的，取决于\({\mathbf {x}}/\varepsilon \)，并且以\(\varepsilon \rightarrow 0\) 的形式快速振荡。为近似值\（E ^ { - tA_ \ varepsilon} \）由 Suslina (Funktsional Analiz i ego Prilozhen 38(4):86–90, 2004) 和 Suslina (Math Model Nat Phenom 5(4):390–447, 2010) 通过光谱方法以及zhikov 和 Pastukhova (Russ J Math Phys 13(2):224–237, 2006) 通过移位方法。在本说明中，我们基于半群的轮廓积分表示和 Suslina (2015) 获得的具有双参数误差估计的解算器的近似值给出了另一个简短的证明。