Spectral approach to homogenization of an elliptic operator periodic in some directions

The operator Aε = D1g1(x1/ε, x2)D1 + D2g2(x1/ε, x2)D2 is considered in L2(R ), where gj(x1, x2), j = 1, 2, are periodic in x1 with period 1, bounded and positive definite. Let function Q(x1, x2) be bounded, positive definite and periodic in x1 with period 1. Let Q(x1, x2) = Q(x1/ε, x2). The behavior of the operator (Aε + Q ) as ε → 0 is studied. It is proved that the operator (Aε +Q ) tends to (A +Q) in the operator norm in L2(R ). Here A is the effective operator whose coefficients depend only on x2, Q 0 is the mean value of Q in x1. A sharp order estimate for the norm of the difference (Aε + Q ) − (A + Q) is obtained. The result is applied to homogenization of the Schrödinger operator with a singular potential periodic in one direction.