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

Abstract

The main result of this work is as follows: for arbitrary pairwise disjoint finite intervals $({\alpha }_j,{\beta }_j)\subset [0,\infty )$, $j=1,\dots ,m$ and for arbitrary $n\ge 2$ we construct the family of periodic non-compact domains $\{{\Omega }^{\epsilon }\subset {\mathbb{R}}^n{\}}_{\epsilon >0}$ such that the spectrum of the Neumann Laplacian in ${\Omega }^{\epsilon }$ has at least $m$ gaps when $\epsilon$ is small enough, moreover the first $m$ gaps tend to the intervals $({\alpha }_j,{\beta }_j)$ as $\epsilon \to 0$. The constructed domain ${\Omega }^{\epsilon }$ is obtained by removing from ${\mathbb{R}}^n$ a system of periodically distributed "trap-like" surfaces. The parameter $\epsilon$ characterizes the period of the domain ${\Omega }^{\epsilon }$, also it is involved in a geometry of the removed surfaces.

23 pages, 2 figures

References in corpus (4)

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• Periodic Riemannian manifold with preassigned gaps in spectrum of Laplace-Beltrami operator
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Cited by in corpus (4)

• An effective Hamiltonian for the eigenvalue asymptotics of a Robin Laplacian with a large parameter
• Gap control by singular Schrödinger operators in a periodically structured metamaterial
• Trapped modes in thin and infinite ladder like domains. Part 1 : existence results
• Gap opening in two-dimensional periodic systems