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Behaviour of zero modes for a one-dimensional Dirac operator arising in models of graphene

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A basic model for conduction within a potential channel in graphene leads naturally to a one-dimensional Dirac operator. The profile of the channel enters the operator as a potential, while zero modes (or zero energy eigenstates) of the operator correspond to conduction modes in the channel.

We consider the behaviour of these zero modes relative to the potential strength and (to a lesser extent) the transversal wave number;both cases can be rephrased as spectral problems for linear operator pencils.

Several results on the eigenvalues of these pencils are presented, in particular relating to their asymptotic distribution.We show that this depends in a subtle way on the sign variation and the presence of gaps or dips in the potential; somewhat more surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential.

This talk is part of the Isaac Newton Institute Seminar Series series.

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