Tuesday, March 12, 2013

1303.2185 (Daniel M. Elton et al.)

Eigenvalues of a one-dimensional Dirac operator pencil    [PDF]

Daniel M. Elton, Michael Levitin, Iosif Polterovich
We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as a spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that the properties of this distribution depend in a subtle way on the sign variation and the presence of gaps in the potential. We also observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented.
View original: http://arxiv.org/abs/1303.2185

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