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Mixed partial-wave scattering with spin-orbit coupling and validity of pseudo-potentials

We present exact solutions of two-body problem for spin-1/2 fermions with isotropic spin-orbit(SO) coupling and interacting with an arbitrary short-range potential. We find that in each partial-wave scattering channel, the parametrization of two-body wavefunction at short inter-particle distance depends on the scattering amplitudes of all channels. This reveals the mixed partial-wave scattering induced by SO couplings. By comparing with results from a square-well potential, we investigate the validity of original pseudo-potential models in the presence of SO coupling. We find the s-wave pseudo-potential provides a good approximation for low-energy solutions near s-wave resonances, given the length scale of SO coupling much longer than the potential range. However, near p-wave resonance the p-wave pseudo-potential gives low-energy solutions that are qualitatively different from exact ones, based on which we conclude that the p-wave model can not be applied to the fermion system if the SO coupling strength is larger or comparable to the Fermi momentum.