In (Formula presented.), consider a second-order elliptic differential operator (Formula presented.), (Formula presented.), of the form (Formula presented.) with periodic coefficients. For small ε, we study the behavior of the semigroup (Formula presented.), t>0, cut by the spectral projection of the operator (Formula presented.) for the interval (Formula presented.). Here (Formula presented.) is the right edge of a spectral gap for the operator (Formula presented.). We obtain approximation for the ‘cut semigroup’ in the operator norm in (Formula presented.) with error (Formula presented.), and also a more accurate approximation with error (Formula presented.) (after singling out the factor (Formula presented.)). The results are applied to homogenization of the Cauchy problem (Formula presented.), (Formula presented.), with the initial data (Formula presented.) from a special class.