Periodic quantum graphs with predefined spectral gaps

Let Γ be an arbitrary periodic metric graph, which does not coincide with a line. We consider the Hamiltonian on its edges; here ϵ > 0 is a small parameter. Let. We show that under a proper choice of vertex conditions the spectrum has at least m gaps as ϵ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of as ϵ → 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) ϵ the precise coincidence of the left endpoints of the first m spectral gaps with predefined numbers.