Dynamics of electrons and explicit solutions of Dirac–Weyl systems

@article{Sakhnovich2016DynamicsOE,
  title={Dynamics of electrons and explicit solutions of Dirac–Weyl systems},
  author={Alexander L. Sakhnovich},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2016},
  volume={50}
}
  • A. Sakhnovich
  • Published 12 September 2016
  • Mathematics
  • Journal of Physics A: Mathematical and Theoretical
Explicit solutions of the Dirac–Weyl system, which are essential in graphene studies, are constructed using our recent approach to the construction of solutions of dynamical systems. The obtained classes of solutions are much wider than the ones which have been considered before. It is proved that neither the constructed potentials nor the corresponding solutions have singularities. Various examples are provided. 
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9 Citations
Background Citations
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9 Citations

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References

SHOWING 1-10 OF 33 REFERENCES
Quasi-exact solution to the Dirac equation for the hyperbolic-secant potential
We analyze bound modes of two-dimensional massless Dirac fermions confined within a hyperbolic secant potential, which provides a good fit for potential profiles of existing top-gated graphene…
Dirac Type System on the Axis: Explicit Formulas for Matrix Potentials with Singularities and Soliton-Positon Interactions
The Jost solutions and the scattering matrix function are constructed explicitly for the Dirac type system on the axis in the case of the rectangular matrix pseudo-exponential potentials. It is shown…
Dynamical canonical systems and their explicit solutions
Dynamical canonical systems and their connections with the classical (spectral) canonical systems are considered. We construct Backlund-Darboux transformation and explicit solutions of the dynamical…
Modified Kortweg-de Vries equation approach to zero-energy states of graphene
We utilize the relation between soliton solutions of the modified Kortweg-de Vries (mKdV) and the combined mKdV-KdV equation and the Dirac equation to construct electrostatic fields which yield exact…
Plasmonics in Dirac systems: from graphene to topological insulators.
  • T. Stauber
  • Physics
    Journal of physics. Condensed matter : an Institute of Physics journal
  • 2014
TLDR
Deep intrinsic plasmon excitation of single and bilayer graphene via hydrodynamic equations and the random phase approximation is discussed and a comprehensive introduction to the standard models and techniques is given.
Dirac electron in graphene under supersymmetry generated magnetic fields
We use supersymmetry transformations to obtain new one parameter families of inhomogeneous magnetic fields B=B˜(x,λ)eˆz?> for which the massless Dirac electron possesses exact solutions. The…
Commutation methods for Schrödinger operators with strongly singular potentials
We explore the connections between singular Weyl–Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in…
Deforming the point spectra of one-dimensional Dirac operators
We provide a method of inserting and removing any finite number of prescribed eigenvalues into spectral gaps of a given one-dimensional Dirac operator. This is done in such a way that the original…
On the integrability of classical spinor models in two-dimensional space-time
Well known classical spinor relativistic-invariant two-dimensional field theory models, including the Gross-Neveu, Vaks-Larkin-Nambu-Jona-Lasinio and some other models, are shown to be integrable by…
Schnol’s Theorem and Spectral Properties of Massless Dirac Operators with Scalar Potentials
The spectra of massless Dirac operators are of essential interest, e.g., for the electronic properties of graphene, but fundamental questions such as the existence of spectral gaps remain open. We…
...
...