Periodic quantum graphs with predefined spectral gaps

Abstract

Abstract Let Γ be an arbitrary Z n -periodic metric graph, which does not coincide with a line. We consider the Hamiltonian H ε on Γ with the action −ɛ −1d2/dx 2 on its edges; here ɛ > 0 is a small parameter. Let m ∈ N . We show that under a proper choice of vertex conditions the spectrum σ ( H ε ) of H ε has at least m gaps as ɛ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of σ ( H ε ) as ɛ → 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) ɛ the precise coincidence of the left endpoints of the first m spectral gaps with predefined numbers.