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First-order Darboux transformations for Dirac equations with arbitrary diagonal potential matrix in two dimensions

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Abstract

We establish first-order Darboux transformations for the two-dimensional Dirac equation with diagonal matrix potential. The potential is allowed to depend arbitrarily on both variables. The systems we consider here include the scenario of a position-dependent mass as well as the massless case. Our Darboux transformations are more general than their existing counterparts (Pozdeeva and Schulze-Halberg in J Math Phys 51:113501, 2010).

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References

  1. A.K. Geim, K.S. Novoselov, The rise of graphene. Nat. Mater. 6, 183 (2007)

    Article  ADS  Google Scholar 

  2. J. Cayssol, Introduction to Dirac materials and topological insulators. Comptes Rendus Physique 14, 760 (2013)

    Article  ADS  Google Scholar 

  3. T.O. Wehling, A.M. Black-Schaffer, A.V. Balatsky, Dirac materials. Adv. Phys. 63, 1 (2014)

    Article  ADS  Google Scholar 

  4. E. McCann, M. Koshino, The electronic properties of bilayer graphene. Rep. Prog. Phys. 76, 056503 (2013)

    Article  ADS  Google Scholar 

  5. E.Y. Andrei, A.H. MacDonald, Graphene bilayers with a Twist. Nat. Mater. 19, 1265 (2020)

    Article  ADS  Google Scholar 

  6. V. Aguiar, S.M. Cunha, D.R. da Costa, R.N. Costa Filho, Dirac fermions in graphene using the position-dependent translation operator formalism. Phys. Rev. B 102, 235404 (2020)

    Article  ADS  Google Scholar 

  7. A. Contreras-Astorga, F. Correa, V. Jakubsky, Super-Klein tunneling of Dirac fermions through electrostatic gratings in graphene. Phys. Rev. B 102, 115429 (2020)

    Article  ADS  Google Scholar 

  8. V. Jakubsky, Spectrally isomorphic Dirac systems: graphene in an electromagnetic field. Phys. Rev. D 91, 045039 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  9. R.R. Hartmann, M.E. Portnoi, Bipolar electron waveguides in graphene. Phys. Rev. B 102, 155421 (2020)

    Article  ADS  Google Scholar 

  10. R.R. Hartmann, M.E. Portnoi, Two-dimensional Dirac particles in a Pöschl-Teller waveguide. Sci. Rep. 7, 11599 (2017)

    Article  ADS  Google Scholar 

  11. M. Chabab, A. El Batoul, H. Hassanabadi, M. Oulne, S. Zare, Scattering states of Dirac particle equation with position dependent mass under the cusp potential. Eur. Phys. J. Plus 131, 387 (2016)

    Article  Google Scholar 

  12. C.L. Ho, P. Roy, Generalized dirac oscillators with position-dependent mass. EPL 124, 60003 (2018)

    Article  ADS  Google Scholar 

  13. M. Erementchouk, P. Mazumder, M.A. Khan, M.N. Leuenberger, Dirac electrons in the presence of matrix potential barrier: application to graphene and topological insulators. J. Phys. Condens. Matter 28, 115501 (2016)

    Article  ADS  Google Scholar 

  14. G. Darboux, Sur une proposition relative aux équations linéaires. C. R. Acad. Sci. 94, 1456–1459 (1882)

    MATH  Google Scholar 

  15. T. Moutard, Sur la construction des equations de la forme \(\frac{1}{z}\frac{d^2z}{dx\;dy}=\lambda (x, y)\) qui admettent une integrale generale explicte. Journal de l’Ecole Polytechnique 45, 1–11 (1878)

  16. T. Moutard, Note sur les equations differentielles lineaires du second ordre. C.R. Acad. Sci. Paris 80, 729–733 (1875)

    MATH  Google Scholar 

  17. C. Gu, A. Hu, Z. Zhou, Darboux Transformations in Integrable Systems (Springer Science and Business Media, Dordrecht, 2005)

    Book  Google Scholar 

  18. V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons (Springer Science and Business Media, Berlin, 1991)

    Book  Google Scholar 

  19. L.M. Nieto, A.A. Pecheritsin, B.F. Samsonov, Intertwining technique for the one-dimensional stationary Dirac equation. Ann. Phys. 305, 151 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  20. G. Junker, Supersymmetric Dirac Hamiltonians in (1+1) dimensions revisited. Eur. Phys. J. Plus 135, 464 (2020)

    Article  Google Scholar 

  21. B. Bagchi, R. Ghosh, Dirac Hamiltonian in a supersymmetric framework. J. Math. Phys. 62, 072101 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  22. E. Pozdeeva, A. Schulze-Halberg, Darboux transformations for a generalized Dirac equation in two dimensions. J. Math. Phys. 51, 113501 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  23. M. Castillo-Celeita, D.J. Fernandez C, Dirac electron in graphene with magnetic fields arising from first-order intertwining operators. J. Phys. A 53, 035302 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  24. A. Contreras-Astorga, D.J. Fernandez C, J. Negro, Solutions of the Dirac equation in a magnetic field and intertwining operators. SIGMA 8, 082 (2012)

    MathSciNet  MATH  Google Scholar 

  25. A. Schulze-Halberg, Higher-order Darboux transformations for the Dirac equation with position-dependent mass at nonvanishing energy. Eur. Phys. J. Plus 135, 863 (2020)

    Article  Google Scholar 

  26. Z. Alizadeh, H. Panahi, Darboux transformations of the one-dimensional stationary Dirac equation with linear potential and its new solutions. Ann. Phys. 409, 167920 (2019)

    Article  MathSciNet  Google Scholar 

  27. A. Sakhnovich, Dynamics of electrons and explicit solutions of Dirac-Weyl systems. J. Phys. A 50, 115201 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  28. N.V. Ustinov, S.B. Leble, Korteweg-de Vries—modified Korteweg-de Vries systems and Darboux transforms in 1+1 and 2+1 dimensions. J. Math. Phys. 34, 1421 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  29. A. Schulze-Halberg, Characterization of Darboux transformations for quantum systems with quadratically energy-dependent potentials, preprint (2021)

  30. A. Schulze-Halberg, Closed-form representations of iterated Darboux transformations for the massless Dirac equation. Int. J. Mod. Phys. A 36, 2150064 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  31. A. Schulze-Halberg, Higher-order Darboux transformations and Wronskian representations for Schrödinger equations with quadratically energy-dependent potentials. J. Math. Phys. 61, 023503 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  32. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1964)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Actuarial Science, and Department of Physics, Indiana University Northwest, 3400 Broadway, Gary, IN, 46408, USA

    Axel Schulze-Halberg

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  1. Axel Schulze-Halberg
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Correspondence to Axel Schulze-Halberg.

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Schulze-Halberg, A. First-order Darboux transformations for Dirac equations with arbitrary diagonal potential matrix in two dimensions. Eur. Phys. J. Plus 136, 790 (2021). https://doi.org/10.1140/epjp/s13360-021-01804-2

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  • DOI: https://doi.org/10.1140/epjp/s13360-021-01804-2

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