On the spectra of periodic elastic beam lattices: Single-layer graph

We consider planar elastic beam Hamiltonians defined on hexagonal lattices. These quantum graphs are constructed from Euler–Bernoulli beams, each governed by the fourth-order Schrödinger operator with a real periodic symmetric potential function. In contrast to the second-order Schrödinger operator commonly studied in the quantum graph literature, here vertex matching conditions encode the geometry of the underlying graph by their dependence on angles at which the edges meet. We show that on the hexagonal lattice, the dispersion relation has a structure similar to that reported for the periodic second-order Schrödinger operator, known as the “graphene Hamiltonian.” This property is then utilized to prove the existence of Dirac points (conical singularities). We further discuss the (ir)reducibility of Fermi surfaces. Moreover, we obtain the point spectrum, the absolutely continuous spectrum, and the singular continuous spectrum. Applying perturbation analysis, we derive the dispersion relation for the planar elastic beam Hamiltonians on angle-perturbed irregular hexagonal lattices, defined in a geometric neighborhood of the hexagonal lattice. On these graphs, we find that, unlike the hexagonal lattice, the dispersion relation is not split into purely energy- and quasimomentum-dependent terms; however, Dirac points exist similar to the hexagonal-lattice case.