On the Spectra of Periodic Elastic Beam Lattices: SingleLayer Graph
Abstract
We present full description of spectra for a Hamiltonian defined on periodic hexagonal elastic lattices. These continua are constructed out of EulerBernoulli beams, each governed by a scalarvalued selfadjoint operator, which is also known as the fourth order Schrödinger operator, equipped with a real periodic symmetric potential. In contrast to the second order Schrödinger operator commonly applied in quantum graph literature, here vertex matching conditions encode geometry of the underlying graph by their dependence on angles at which edges are met. We show that for a special equal angle lattice, known as graphene, dispersion relation has a similar structure as reported for the periodic second order Schrödinger operator on hexagonal lattices. This property is then further utilized to prove existence of singular Dirac points. We further discuss reducibility of Fermi surface at uncountably many lowenergy levels for this special lattice. Applying perturbation analysis, we extend the developed theory to derive dispersion relation for angleperturbed Hamiltonian of hexagonal lattices in a geometric neighborhood of graphene. In these graphs, unlike graphene, dispersion relation is not splitted into purely energy and quasimomentum dependent terms, however singular Dirac points exist similar to 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
 EPrint:
 arXiv admin note: text overlap with arXiv:mathph/0612021 by other authors