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In, consider a second-order elliptic differential operator,, of the formwith periodic coefficients. For small  ε , we study the behavior of the semigroup,  t &gt;0, cut by the spectral projection of the operatorfor the interval. Hereis the right edge of a spectral gap for the operator. We obtain approximation for the ‘cut semigroup’ in the operator norm inwith error, and also a more accurate approximation with error(after singling out the factor). The results are applied to homogenization of the Cauchy problem,, with the initial datafrom a special class.

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