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The operatorvboxis considered in  L  2 (ℝ 2 ), where  g   j  ( x  1 ,  x  2 ),  j  = 1, 2, are periodic in  x  1  with period 1, bounded and positive definite. Let function  Q ( x  1 ,  x  2 ) be bounded, positive definite and periodic in  x  1  with period 1. Let  Q  ε ( x  1 ,  x  2 ) =  Q ( x  1 /ε,  x  2 ). The behavior of the operator ( A  ε  +  Q  ε ) −1  as ε→0 is studied. It is proved that the operator ( A  ε  +  Q  ε ) −1  tends to ( A  0  +  Q  0 ) −1  in the operator norm in  L  2 (ℝ 2 ). Here,  A  0  is the effective operator whose coefficients depend only on  x  2 ,  Q  0  is the mean value of  Q  in  x  1 . A sharp order estimate for the norm of the difference ( A  ε  +  Q  ε ) −1 −( A  0  +  Q  0 ) −1  is obtained. The result is applied to homogenization of the Schrödinger operator with a singular potential periodic in one direction. Copyright © 2011 John Wiley & Sons, Ltd.

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