Let \(A_{\varepsilon } = \textbf{D}^* g(\varvec{x}/\varepsilon ) \textbf{D} + \varepsilon ^{-2} p({\varvec{x}}/\varepsilon ),\) \(\varepsilon >0\) , be a second-order elliptic differential operator in \(L_2(\mathbb {R}^d)\) with periodic coefficients. For small \(\varepsilon \) , the behavior of the semigroup \(e^{- A_{\varepsilon } t}\) , \(t>0\) , cut by the spectral projection of \(A_{\varepsilon }\) into the interval \([\varepsilon ^{-2} \lambda _{+},+\infty )\) is studied. Here, \(\varepsilon ^{-2} \lambda _{+}\) is the right edge of the spectral gap for \(A_{\varepsilon }\) . An approximation for the “cut semigroup” in the operator norm on \(L_2(\mathbb {R}^d)\) with error \(O(\varepsilon )\) is obtained together with a more accurate approximation with corrector taken into account with error \(O(\varepsilon ^2)\) (after singling out the factor \(e^{-t \lambda _{+} / \varepsilon ^2}\) ). The results are applied to homogenization of the Cauchy problem \(\partial _t v_\varepsilon = - A_\varepsilon v_\varepsilon \) , \(v_\varepsilon \vert _{t=0} = f_\varepsilon \) , with initial data \(f_\varepsilon \) from a special class. Bibliography: 24 titles.
