Nondivergent pseudopotential treatment of spin-polarized fermions under one- and three-dimensional harmonic confinement

Atom-atom scattering of bosonic one-dimensional (1D) atoms has been modeled successfully using a zero-range δ-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.