OPERATOR ERROR ESTIMATES IN THE HOMOGENIZATION PROBLEM FOR NONSTATIONARY PERIODIC EQUATIONS

Matrix periodic differential operators (DO's) A = A(x, D) in L(2)(R(d); C(n)) are considered. The operators are assumed to admit a factorization of the form A = chi*chi, where chi is a homogeneous first order DO. Let A(epsilon) = A(epsilon(-1)x, D), epsilon > 0. The behavior of the solutions u(epsilon)(x, tau) of the Cauchy problem for the Schrodinger equation i partial derivative(tau)u(epsilon) = A(epsilon)u(epsilon), and also the behavior of those for the hyperbolic equation partial derivative(2)(tau)u(epsilon) = -A(epsilon)u(epsilon) is studied as epsilon -> 0. Let u(0) be the solution of the corresponding homogenized problem. Estimates of order epsilon are obtained for the L(2)(R(d); C(n))-norm of the difference u(epsilon) - u(0) for a fixed tau is an element of R. The estimates are uniform with respect to the norm of initial data in the Sobolev space H(s)(R(d); C(n)), where s = 3 in the case of the Schrodinger equation and s = 2 in the case of the hyperbolic equation. The dependence of the constants in estimates on the time tau is traced, which makes it possible to obtain qualified error estimates for small E and large vertical bar tau vertical bar = O(epsilon(-alpha)) with appropriate alpha < 1.