Homogenization of the multidimensional parabolic equations with periodic coefficients at the edge of a spectral gap

In $ L_2(\mathbb{R}^d) $, we consider a second-order elliptic differential operator $A_{\varepsilon} = \mathbf{D}^* g(\mathbf{x}/\varepsilon) \mathbf{D} + \varepsilon^{-2}p(\mathbf{x}/\varepsilon),$ $ \varepsilon > 0 $, with periodic coefficients. For small $ \varepsilon $, we study the behavior of the semigroup $ e^{-A_{\varepsilon}t} $, $ t > 0 $, cut by the spectral projection of the operator $ A_{\varepsilon} $ for the interval $ [\varepsilon^{-2}\lambda_{+}, +\infty) $. Here $ \varepsilon^{-2}\lambda_{+} $ is the right edge of a spectral gap for the operator $ A_{\varepsilon} $. We obtain approximation for the 'cut semigroup' in the operator norm in $L_2(\mathbb{R}^d)$ with error $O(\varepsilon)$, and also a more accurate approximation 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 the initial data $f_\varepsilon$ from a special class.