In one dimension the integration is straightforward. From (7.29) we find
![]() |
(7.38) |
The integration over gives
if
and zero otherwise. So, the force is
![]() |
(7.39) |
For the non-resonance scattering
, where
. The expression of the force
in terms of the scattering length reads as
An interesting peculiarity is that the result does not depend on the velocity
(where, of course, the velocity must be larger than the speed of sound). This
phenomenon comes from particular properties of a
-potential, namely that the
Fourier transformation of this potential is a constant. Numerical solutions by
Pavloff[Pav02] for finite-range potentials in
show no friction for
, maximal friction for
and smaller friction for
, although the
constant result (7.41) was found for the
-potential.
In a 1D system energy dissipation is possible at due to creation of the ``gray
solitons'' first considered in [Tsu71]. Non-linear calculations [Hak97]
show that the critical velocity for this process decreases with increasing coupling
constant
.
This theory can be checked in an experiment in a three-dimensional condensate. The impurity can be presented by a moving light sheet.