Abstract. We deal with operators in R n of the formA = −1b(x)X nk=1 ∂∂x k a(x)∂∂x k !where a(x),b(x) are positive, bounded and periodic functions. We denote by L per the set of such operators.The main result of this work is as follows: for an arbitrary L >0 and for arbitrary pairwise disjoint intervals(α j ,β j ) ⊂ [0,L], j = 1,...,m (m ∈ N) we construct the family of operators {A e ∈ L per } e such that thespectrum of A e has exactly m gaps in [0,L] when eis small enough, and these gaps tend to the intervals(α j ,β j ) as e→ 0. The idea how to construct the family {A e } e is based on methods of the homogenizationtheory.Keywords: periodic elliptic operators, spectrum, gaps, homogenization. IntroductionOur research is inspired by the following well-known resultof Y. Colin de Verdie`re [4]: forarbitrary numbers 0 = λ 1 <λ 2 <··· <λ m (m ∈ N) and n ∈ N {1} there is a n-dimensionalcompact Riemannian manifold M such that the first m eigenvalues of the corresponding Laplace-Beltrami operator −∆