  We calculate effects of an applied helically symmetric potential on the low
energy electronic spectrum of a carbon nanotube in the continuum approximation.
The spectrum depends on the strength of this potential and on a dimensionless
geometrical parameter, P, which is the ratio of the circumference of the
nanotube to the pitch of the helix. We find that the minimum band gap of a
semiconducting nanotube is reduced by an arbitrarily weak helical potential,
and for a given field strength there is an optimal P which produces the biggest
change in the band gap. For metallic nanotubes the Fermi velocity is reduced by
this potential and for strong fields two small gaps appear at the Fermi surface
in addition to the gapless Dirac point. A simple model is developed to estimate
the magnitude of the field strength and its effect on DNA-CNT complexes in an
aqueous solution. We find that under typical experimental conditions the
predicted effects of a helical potential are likely to be small and we discuss
several methods for increasing the size of these effects.
