2012/01/18 by Andrii Khrabustovskyi, Khrabustovskyi, Andrii
Mathematics · Physics and Astronomy · #35P20 #35P25 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.AP #math.MP #math.SP #msc:35P20 #msc:35P25
paper · pdf · doi:10.48550/arxiv.1201.3729
28 pages, 2 figures
arxiv created 2012/02/03 · arxiv updated 2012/02/06
We deal with operators in ℝn of the form A=-1\over b(x)∑k=1n\ds∂\over∂ xk(a(x)∂ \over∂ 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 A\e_\eps is based on methods of the homogenization theory.