Let Aε=D∗g(x/ε)D+ε-2p(x/ε),ε>0, be a second-order elliptic differential operator in L2(Rd) with periodic coefficients. For small ε, the behavior of the semigroup e-Aεt, t>0, cut by the spectral projection of Aε into the interval [ε-2λ+,+∞) is studied. Here, ε-2λ+ is the right edge of the spectral gap for Aε. An approximation for the “cut semigroup” in the operator norm on L2(Rd) with error O(ε) is obtained together with a more accurate approximation with corrector taken into account with error O(ε2) (after singling out the factor e-tλ+/ε2). The results are applied to homogenization of the Cauchy problem ∂tvε=-Aεvε, vε|t=0=fε, with initial data fε from a special class. Bibliography: 24 titles. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.