The spectra of rectangular lattices of quantum waveguides

We obtain asymptotic formulae for the spectral segments of a thin ($h\ll 1$) rectangular lattice of quantum waveguides which is described by a Dirichlet problem for the Laplacian. We establish that the structure of the spectrum of the lattice is incorrectly described by the commonly accepted quantum graph model with the traditional Kirchhoff conditions at the vertices. It turns out that the lengths of the spectral segments are infinitesimals of order $O(e^{-\delta/h})$, $\delta>0$, and $O(h)$ as $h\to+0$, and gaps of width $O(h^{-2})$ and $O(1)$ arise between them in the low-frequency and middle-frequency spectral ranges respectively. The first spectral segment is generated by the (unique) eigenvalue in the discrete spectrum of an infinite cross-shaped waveguide $\Theta$. The absence of bounded solutions of the problem in $\Theta$ at the threshold frequency means that the correct model of the lattice is a graph with Dirichlet conditions at the vertices which splits into two infinite subsets of identical edges-intervals. By using perturbations of finitely many joints, we construct any given number of discrete spectrum points of the lattice below the essential spectrum as well as inside the gaps.