J. Brüning, S. Yu. Dobrokhotov, M. I. Katsnel'son, D. S. Minenkov, “Semiclassical asymptotic approximations and the density of states for the two-dimensional radially symmetric Schrödinger and Dirac equations in tunnel microscopy problems”, TMF, 2016, Volume 186, Number 3,Pages <nobr>386

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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

J. Brüning^a, S. Yu. Dobrokhotov^bc, M. I. Katsnel'son^de, D. S. Minenkov^bc

^a Humboldt University, Berlin, Germany
^b Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region, Russian
^c Institute for Problems in Mechanics, RAS, Moscow, Russia
^d Institute for Molecules and Materials, Radboud University, Nijmegen, the Netherlands
^e Ural Federal University, Ekaterinburg, Russia

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.

Keywords: axially symmetric two-dimensional Schrödinger operator, axially symmetric two-dimensional Dirac operator, generalized eigenfunction, semiclassical approximation, density of states, tunnel microscopy.

Received: 10.06.2015

DOI: 10.4213/tmf8987