The authors focus on the Floquet-Bloch spectral theory of the Schrödinger operator \[ H^{(\varepsilon)} \equiv -\Delta + \varepsilon V_h(\mathbf{x}),\quad\mathbf{x} \in \mathbb{R}^2, \] where \(V_h\) is a \(C^{\infty}\) periodic potential having honeycomb structure symmetry and \(\varepsilon\) is a real parameter, not necessarily small in absolute value. A consequence of the honeycomb symmetry is that the first Brillouin zone \(\mathcal{B}\) is a regular hexagon.
The authors prove (Theorem 5.1) that the dispersion surface of \(H^{(\varepsilon)}\mathfrak{}\) has conical singularities at each vertex of \(\mathcal{B}\), except possibly for a discrete set of \(\varepsilon\)’s. They, also, show (Theorem 9.1) that conical singularities persist for perturbed potentials, which are even but not necessarily honeycomb-symmetric. In this case, the conical singularities, typically, do not occur at the vertices of \(\mathcal{B}\).