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

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.