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Matrix periodic elliptic second order differential operators Bε in Rd with rapidly oscillating coefficients (depending on x/ε) are studied. The principal part of the operator is given in a factorized form b(D)∗g(ε−1x)b(D), where g is a periodic, bounded and positive definite matrix-valued function and b(D) is a matrix first order operator whose symbol is a matrix of maximal rank. The operator also has zero and first order terms with unbounded coefficients. The problem of homogenization in the small period limit is considered. Approximation for the generalized resolvent of the operator Bε is obtained in the operator norm in L2(R;C) with error term O(ε). Also, approximation for this resolvent is obtained in the norm of operators acting from L2(R;C) to H1(Rd;Cn) with error term of order ε and with the corrector taken into account. The general results are applied to homogenization problems for the Schrödinger operator and the two-dimensional Pauli operator with potentials involving singular terms. §0. Introduction 0.1. Main results. We study the problem of homogenization in the small period limit for matrix differential operators (DO’s) Bε, ε > 0, acting in the space L2(R;C). Let Γ be a lattice in R, and let Ω be the cell of Γ. We use the notation φ(x) = φ(ε−1x) for any measurable Γ-periodic function φ(x) in R. The principal part Aε of the operator Bε is given in a factorized form Aε = b(D)∗gε(x)b(D), where b(D) is a matrix homogeneous first order DO and g(x) is a bounded and positive definite matrix-valued function in R, periodic with respect to the lattice Γ. (The precise assumptions on b(D) and g(x) are given below in §5.) Homogenization problems for the operator Aε have been studied in detail in a series of papers [BSu1, BSu2, BSu3, BSu4, BSu5]. Now we study more general operators Bε: we add zero and first order terms to Aε: (0.1) Bεu = Aεu+ d ∑ j=1 (aj(x)Dju+Dj(a ε j(x)) ∗u) +Q(x)u. Here the aj(x), j = 1, . . . , d, are Γ-periodic (n × n)-matrix-valued functions such that aj ∈ Lρ(Ω), ρ = 2 for d = 1, and ρ > d for d ≥ 2. In general, the potential Q(x) is a distribution (with values in the class of Hermitian matrices) generated by a rapidly oscillating matrix-valued measure. It is only assumed that the generalized potential Q 2010 Mathematics Subject Classification. Primary 35B27.

Homogenization in the Sobolev class $H^{1}(\mathbb R^{d})$ for second order periodic elliptic operators with the inclusion of first order terms