On the Spectra of Periodic Elastic Beam Lattices: SingleLayer Graph
Abstract
We present full description of spectra for elastic beam Hamiltonian defined on periodic hexagonal lattices. These continua are constructed out of EulerBernoulli beams, each governed by a scalar valued selfadjoint fourthorder operator equipped with a real periodic symmetric potential. Compared to the Schrödinger operator commonly applied in quantum graph literature, here vertex matching conditions encode geometry of the graph by their dependence on angles at which edges are met. We show that for a special equalangle lattice, known as graphene, dispersion relation has a similar structure as reported for Schrödinger operator on periodic hexagonal lattices. This property is then further utilized to prove existence of singular Dirac points. We next discuss the role of the potential on reducibility of Fermi surface at uncountably many lowenergy levels for this special lattice. Applying perturbation analysis, the developed theory is extended to derive dispersion relation for angleperturbed Hamiltonian of lattices in a geometricneighborhood of graphene. In these graphs, unlike graphene, dispersion relation is not splitted into purely energy and quasimomentum dependent terms, but up to some quantifiable accuracy, singular Dirac points exist at the same points as the graphene case.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.05466
 Bibcode:
 2021arXiv211005466E
 Keywords:

 Mathematical Physics;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Spectral Theory;
 05C90;
 58J50;
 35J50;
 34L40;
 47A75;
 47B25