Periodic elliptic operators with asymptotically preassigned spectrum

Abstract

We deal with operators in Rⁿ of the form

A=−1/b(x)Σ_k=1ⁿ∂/∂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^ε∈L_per}_ε such that the spectrum of A^ε has exactly m gaps in [0,L] when ε is small enough, and these gaps tend to the intervals (α_j,β_j) as ε→0. The idea how to construct the family {A^ε}_ε is based on methods of the homogenization theory.