Periodic quantum graphs with predefined spectral gaps

Abstract

Let $\Gamma$ be an arbitrary ${\mathbb{Z}}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian ${\mathcal{H}}_{\epsilon }$ on $\Gamma$ with the action $-{\epsilon }^{-1}{\mathrm{d}}^2/\mathrm{d}{x}^2$ on its edges; here $\epsilon >0$ is a small parameter. Let $m\in \mathbb{N}$. We show that under a proper choice of vertex conditions the spectrum $\sigma ({\mathcal{H}}^{\epsilon })$ of ${\mathcal{H}}^{\epsilon }$ has at least $m$ gaps as $\epsilon$ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of $\sigma ({\mathcal{H}}^{\epsilon })$ as $\epsilon \to 0$ can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) $\epsilon$ the precise coincidence of the left endpoints of the first $m$ spectral gaps with predefined numbers.

18 pages, 4 figures