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

In L-2(R), consider a second-order elliptic differential operator A(epsilon), epsilon > 0, of the form A(epsilon) = - d dx g(x/epsilon) d/dx + epsilon-2p(x/epsilon) with periodic coefficients. For small epsilon, we study the behavior of the semigroup e-Ae t, t> 0, cut by the spectral projection of the operator Ae for the interval [epsilon -2.,+8). Here e-2. is the right edge of a spectral gap for the operator A(epsilon). We obtain approximation for the `cut semigroup' in the operator norm in L2( R) with error O(epsilon), and also a more accurate approximation with error O(epsilon 2) (after singling out the factor e-t./epsilon(2)). The results are applied to homogenization of the Cauchy problem.tve = -A(epsilon)ve, ve | t=0 = f(epsilon), with the initial data fe from a special class.