  The recently discovered soft gamma-ray emission from the anomalous X-ray
pulsar 1E 1841-045 has a luminosity L_g ~ 10^{36} ergs/s. This luminosity
exceeds the spindown power by three orders of magnitude and must be fed by an
alternative source of energy such as an ultrastrong magnetic field. A gradual
release of energy in the stellar magnetosphere is expected if it is twisted and
a strong electric current is induced on the closed field lines. We examine two
mechanisms of gamma-ray emission associated with the gradual dissipation of
this current. (1) A thin surface layer of the star is heated by the downward
beam of current-carrying charges, which excite Langmuir turbulence in the
layer. As a result, it can reach a temperature kT ~ 100 keV and emit
bremsstrahlung photons up to this characteristic energy. (2) The magnetosphere
is also a source of soft gamma rays at a distance of ~100 km from the star,
where the electron cyclotron energy is in the keV range. A large electric field
develops in this region in response to the outward drag force felt by the
current-carrying electrons from the flux of keV photons leaving the star. A
seed positron injected in this region undergoes a runaway acceleration and
upscatters keV photons above the threshold for pair creation. The created pairs
emit a synchrotron spectrum consistent with the observed 20-100 keV emission.
This spectrum is predicted to extend to higher energies and reach a peak at ~1
MeV.
