Spectrum of a dilated honeycomb network (Q2342942)

The authors consider a honeycomb network, i.e., an infinite, periodic metric graph \(\Gamma\) whose basic cell is a hexagon with equal antipodal edge lengths. The object of interest is the selfadjoint Laplacian \(H\) in \(L^2 (\Gamma)\) subject to a \(\delta\)-coupling of strength \(\alpha \in \mathbb{R}\) at each vertex. A detailed study of the spectral properties of \(H\) is provided. Amongst others the authors prove that eigenvalues exist if and only if the three different appearing edge lengths \(a, b, c\) are commensurate and that in the noncommensurate case the spectrum of \(H\) is purely absolutely continuous. Moreover, conditions for the existence of spectral gaps are shown in terms of the edge lenghts \(a, b, c\) and the strength \(\alpha\) of the interaction. For the special case where the edge lengths \(b\) and \(c\) coincide these conditions are specified further in terms of number theoretic properties of the ratio \(a/b\).