Opening up and control of spectral gaps of the Laplacian in periodic domains

The main result of this work is as follows: for arbitrary pairwise disjoint, finite intervals (αj, βj) ⊂ [0, ∞), j = 1, …, m, and for arbitrary n ≥ 2, we construct a family of periodic non-compact domains {Ωε⊂ℝn}ε>0 such that the spectrum of the Neumann Laplacian in Ωε has at least m gaps when ε is small enough, moreover the first m gaps tend to the intervals (αj, βj) as ε → 0. The constructed domain Ωε is obtained by removing from ℝn a system of periodically distributed “trap-like” surfaces. The parameter ε characterizes the period of the domain Ωε, also it is involved in a geometry of the removed surfaces.