Periodic quantum graphs with predefined spectral gaps

Abstract

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