Abstract
In this article, we analyze the spectrum of discrete magnetic Laplacians (DML) on an infinite covering graph G ˜ → G = G ˜ / Γ with (Abelian) lattice group Γ and periodic magnetic potential β ˜ . We give sufficient conditions for the existence of spectral gaps in the spectrum of the DML and study how these depend on β ˜ . The magnetic potential can be interpreted as a control parameter for the spectral bands and gaps. We apply these results to describe the spectral band/gap structure of polymers (polyacetylene) and nanoribbons in the presence of a constant magnetic field.
Highlights
It is a well-known fact that the spectrum of Laplacians or, more generally, Schrödinger operators with periodic potentials, on Abelian coverings, have band structure
If two consecutive spectral bands of a bounded self-adjoint operator T do not overlap, we say that the spectrum has a spectral gap, i.e., a maximal nonempty interval ( a, b) ⊂ [−k T k, k T k] that does not intersect the spectrum of the operator
We study the spectrum of discrete magnetic Laplacians (DMLs for short) on infinite discrete coverings graphs e → G = G/Γ
Summary
It is a well-known fact that the spectrum of Laplacians or, more generally, Schrödinger operators with periodic potentials, on Abelian coverings, have band structure. To show the existence of spectral gaps, we develop a purely discrete spectral f β localization technique based on the virtualization of arcs and vertices on quotient G These operations produce new graphs with, in general, different weights that allow localizing the eigenvalues of the original Laplacian inside certain intervals. It can be seen how a periodic magnetic potential with constant value βe ∈ [0, 2π ) on each cycle (and plotted on the horizontal axis) affects the spectral bands (gray vertical intervals that appear as the intersection of the region with a line βe = const) and the spectral gaps (white vertical intervals).
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