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Semiclassical asymptotic approximations and the density of states for the two-dimensional radially symmetric Schrödinger and Dirac equations in tunnel microscopy problems

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Abstract

We consider the two-dimensional stationary Schrödinger and Dirac equations in the case of radial symmetry. A radially symmetric potential simulates the tip of a scanning tunneling microscope. We construct semiclassical asymptotic forms for generalized eigenfunctions and study the local density of states that corresponds to the microscope measurements. We show that in the case of the Dirac equation, the tip distorts the measured density of states for all energies.

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Authors and Affiliations

  1. Humboldt University, Berlin, Germany

    J. Brüning

  2. Institute for Problems in Mechanics, RAS, Moscow, Russia

    S. Yu. Dobrokhotov & D. S. Minenkov

  3. Moscow Institute of Physics and Technology, Moscow, Russia

    S. Yu. Dobrokhotov

  4. Institute for Molecules and Materials, Radboud University, Nijmegen, the Netherlands

    M. I. Katsnelson

  5. Ural Federal University, Ekaterinburg, Russia

    M. I. Katsnelson

Authors
  1. J. Brüning
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  2. S. Yu. Dobrokhotov
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  3. M. I. Katsnelson
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  4. D. S. Minenkov
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Corresponding author

Correspondence to J. Brüning.

Additional information

This research was supported by the DFG (Grant No. SFB 647/3), the Russian Foundation for Basic Research (Grant No. 14-01-00521), and the Russian Federation (Act 211, Contract No. 02.A03.21.0006).

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 186, No. 3, pp. 386–400, March, 2016.

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Brüning, J., Dobrokhotov, S.Y., Katsnelson, M.I. et al. Semiclassical asymptotic approximations and the density of states for the two-dimensional radially symmetric Schrödinger and Dirac equations in tunnel microscopy problems. Theor Math Phys 186, 333–345 (2016). https://doi.org/10.1134/S004057791603003X

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  • DOI: https://doi.org/10.1134/S004057791603003X

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