Periodic elliptic operators with asymptotically preassigned spectrum

Article type: Research Article

Authors: Khrabustovskyi, Andrii

Affiliations: Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Lenin avenue 47, Kharkiv 61103, Ukraine. Tel.: +38 057 3410986; E-mail: andry9@ukr.net

Abstract: We deal with operators in Rn of the form A=−1/b(x)Σk=1n∂/∂xk(a(x)∂/∂xk), where a(x),b(x) are positive, bounded and periodic functions. We denote by Lper 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 (αj,βj)⊂[0,L], j=1,…,m (m∈N) we construct the family of operators {Aε∈Lper}ε such that the spectrum of Aε has exactly m gaps in [0,L] when ε is small enough, and these gaps tend to the intervals (αj,βj) as ε→0. The idea how to construct the family {Aε}ε is based on methods of the homogenization theory.

Keywords: periodic elliptic operators, spectrum, gaps, homogenization

DOI: 10.3233/ASY-2012-1131

Journal: Asymptotic Analysis, vol. 82, no. 1-2, pp. 1-37, 2013