On the graphene Hamiltonian operator
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We solve a second-order elliptic equation with quasi-periodic boundary conditions defined on a honeycomb lattice that represents the arrangement of carbon atoms in graphene. Our results generalize those found by Kuchment and Post (Commun Math Phys 275(3):805–826, 2007) to characterize not only the stability but also the instability intervals of the solutions. This characterization is obtained from the solutions of the energy eigenvalue problem given by the lattice Hamiltonian. We employ tools of the one-dimensional Floquet theory and specify under which conditions the one-dimensional theory is applicable to the structure of graphene. The systematic study of such stability and instability regions provides a tool to understand the propagation properties and behavior of the electrons wavefunction in a hexagonal lattice, a key problem in graphene-based technologies.
KeywordsPeriodic solutions General spectral theory Spectral theory and eigenvalue problems Graphene Honeycomb structure
Mathematics Subject Classification34L05 82D80 34B60 34B45 47A10 34D20
C. Conca and J. San Martín were partially supported from PFBasal-01 (CeBiB), PFBasal-03 (CMM) projects. C.Conca also received partial support from Ecos-Conicyt Grant C13E05 and by Fondecyt Grant 1140773. J. San Martín also received partial support from Fondecyt Grant 1180781. V. Solano was partially supported by Scholarship Program of CONICYT-Chile, Folio Number 21110749, by PFBasal-03 (CMM) project and by the Grants SEV-2011-0087 from Ministerio de Ciencia e Innovación (MICINN) of Spain. We would like to thank M. Solano for valuable comments on the manuscript.
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