  We study the relaxation of the center-of-mass, or dipole oscillations in the
system of interacting fermions confined spatially. With the confinement
frequency $\omega_{\perp}$ fixed the particles were considered to freely move
along one (quasi-1D) or two (quasi-2D) spatial dimensions. We have focused on
the regime of rare collisions, such that the inelastic collision rate,
$1/\tau_{in} \ll \omega_{\perp}$. The dipole oscillations relaxation rate,
$1/\tau_{\perp}$ is obtained at three different levels: by direct perturbation
theory, solving the integral Bethe-Salpeter equation and applying the memory
function formalism. As long as anharmonicity is weak, $1/\tau_{\perp} \ll 1/
\tau_{in}$ the three methods are shown to give identical results. In quasi-2D
case $1/\tau_{\perp} \neq 0$ at zero temperature. In quasi-1D system
$1/\tau_{\perp} \propto T^3$ if the Fermi energy, $E_F$ lies below the critical
value, $E_F < 3 \omega_{\perp}/4$. Otherwise, unless the system is close to
integrability, the rate $1/\tau_{\perp}$ has the temperature dependence similar
to that in quasi-2D. In all cases the relaxation results from the excitation of
particle-hole pairs propagating along unconfined directions resulting in the
relationship $1/\tau_{\perp} \propto 1/\tau_{in}$, with the inelastic rate
$1/\tau_{in} \neq 0$ as the phase-space opens up at finite energy of
excitation, $\hbar \omega_{\perp}$. While $1/\tau_{\perp} \propto \tau_{in}$ in
the hydrodynamic regime, $\omega_{\perp} \ll 1/\tau_{in}$, in the regime of
rare collisions, $\omega_{\perp} \gg 1/\tau_{in}$, we obtain the opposite trend
$1/\tau_{\perp} \propto 1/\tau_{in}$.
