In $L_2\left(\mathbb{R}\right)$, consider a second-order elliptic differential operator $A_{ϵ}$, $ϵ>0$, of the form $A_{ϵ}=-\frac{d}{dx}g\left(x/ϵ\right)\frac{d}{dx}+{ϵ}^{-2}p\left(x/ϵ\right)$ with periodic coefficients. For small ε, we study the behavior of the semigroup $e^{-A_{ϵ}t}$, t>0, cut by the spectral projection of the operator $A_{ϵ}$ for the interval $[{ϵ}^{-2}\nu ,+\mathrm{\infty })$. Here ${ϵ}^{-2}\nu$ is the right edge of a spectral gap for the operator $A_{ϵ}$. We obtain approximation for the ‘cut semigroup’ in the operator norm in $L_2\left(\mathbb{R}\right)$ with error $O\left(ϵ\right)$, and also a more accurate approximation with error $O\left({ϵ}^2\right)$ (after singling out the factor $e^{-t\nu /{ϵ}^2}$). The results are applied to homogenization of the Cauchy problem ${\mathrm{\partial }}_t{v}_{ϵ}=-A_{ϵ}{v}_{ϵ}$, $v_{ϵ}{|}_{t=0}=f_{ϵ}$, with the initial data $f_{ϵ}$ from a special class.

在${\mathrm{大号}}_2\left(\mathbb{R}\right)$, 考虑一个二阶椭圆微分算子${\mathrm{一种}}_{\epsilon }$,$\epsilon >0$, 形式${\mathrm{一种}}_{\epsilon }=-\frac{d}{dX}G\left(X/\epsilon \right)\frac{d}{dX}+{\epsilon }^{-2}p\left(X/\epsilon \right)$具有周期性系数。对于小的ε，我们研究半群的行为$e^{-{\mathrm{一种}}_{\epsilon }吨}$, t >0, 被算子的光谱投影切割${\mathrm{一种}}_{\epsilon }$对于间隔$[{\epsilon }^{-2}\nu ,+\mathrm{\infty })$. 这里${\epsilon }^{-2}\nu$是算子光谱间隙的右边缘${\mathrm{一种}}_{\epsilon }$. 我们在算子范数中获得了“截半群”的近似值${\mathrm{大号}}_2\left(\mathbb{R}\right)$有错误$○\left(\epsilon \right)$, 也是一个更准确的近似误差$○\left({\epsilon }^2\right)$（在挑选出因素后$e^{-吨\nu /{\epsilon }^2}$）。结果应用于柯西问题的同质化${\mathrm{\partial }}_{吨}{v}_{\epsilon }=-{\mathrm{一种}}_{\epsilon }{v}_{\epsilon }$,$v_{\epsilon }{|}_{吨=0}=F_{\epsilon }$, 与初始数据$F_{\epsilon }$从一个特殊的班级。