We elucidate the physics underlying the fact that both bright and dark solitary waves can arise in a one-dimensional spin-polarized gas of fermionic atoms with attractive three-dimensional
p-wave interactions in a hard-wall trap. This is possible since the one-dimensional fermion system can be mapped to a system of bosons described by the Lieb-Linger model with either repulsive or attractive $\ensuremath{\delta}$-function interactions which can support solitary waves.
Atom-atom scattering of bosonic one-dimensional (1D) atoms has been modeled successfully using a zero-range $\ensuremath{\delta}$-function potential, while that of bosonic 3D atoms has been modeled successfully using Fermi-Huang's regularized
s-wave pseudopotential. Here, we derive the eigenenergies of two spin-polarized 1D fermions under external harmonic confinement interacting through a zero-range potential, which only acts on odd-parity wave functions, analytically. We also present a divergent-free zero-range potential treatment of two spin-polarized 3D fermions under harmonic confinement. Our pseudopotential treatments are verified through numerical calculations for short-range model potentials.
Spin propagation in systems of one-dimensional interacting fermions at finite temperature is intrinsically diffusive. The spreading rate of a spin packet is controlled by a transport coefficient termed ``spin drag'' relaxation time ${\ensuremath{\tau}}_{\mathrm{sd}}$. In this paper we present both numerical and analytical calculations of ${\ensuremath{\tau}}_{\mathrm{sd}}$ for a two-component spin-polarized cold Fermi gas trapped inside a tight atomic waveguide. At low temperatures we find an activation law for ${\ensuremath{\tau}}_{\mathrm{sd}}$, in agreement with earlier calculations of Coulomb drag between slightly asymmetric quantum wires, but with a different and much stronger temperature dependence of the prefactor. Our results provide a fundamental input for microscopic time-dependent spin-density functional theory calculations of spin transport in one-dimensional inhomogeneous systems of interacting fermions.