  We consider periodically driven potential impurities coupled to the surface
states of a two-dimensional topological insulator. The problem is addressed by
means of two models, out which the first model is an effective continuum
Hamiltonian for the surface states, whereas the Kane-Mele lattice model is our
second approach. While both models result in drastic changes in the local
density of electron states with increasing amplitude and frequency of the
driving field, the linearly low energy local density of electron states remains
in the continuum model, however, with an increased Fermi velocity. The spectrum
of the continuum model remains gapless under the emergence of new impurity
resonances near the Fermi energy. The Kane-Mele lattice model represents a
finite size system, with edge states appearing at the boundary of the system.
We, thus, consider the impurity at two different positions, one at the boundary
and one at the center of the lattice. In the former case, a reduction and
broadening of the low energy local density of electron states result with
increasing amplitude of the driving field. On the other hand, there are no new
resonances emerging in the spectrum. In the latter case, the spectrum is gapped
both in the absence of the impurity as well as for weak amplitudes of the
driving field, while the gap tends to fill up with impurity states with
increasing amplitude.
