  Motivated by recent progresses on ultracold alkaline-earth atoms towards the
goal of simulating Kondo physics, in this work we exactly solve the few-body
problem of one and two trapped fermions in one dimension interacting with a
localized impurity under tunable spin-exchange interaction. It is found that
depending on the sign of the spin-exchange coupling, ferromagnetic(FM) or
anti-ferromagnetic(AFM), the attractive and repulsive branches can hold
different magnetic structures. For the two fermions case, we demonstrate the
Kondo screening effect for the attractive branch with AFM coupling, and show
that such screening is absent for the ground state with FM coupling. Moreover,
we find a sequence of FM upper branches in the AFM coupling side. These FM
states are orthogonal to all other attractive branches and their wave functions
feature a full spin-charge separation. The effect of an additional contact
interaction and the extension of our results to many particles are also
discussed. This work reveals the intriguing physics uniquely associated with
the spin-exchange interaction in the few-body point of view, which are
promisingly to be explored in the experiment of ultracold alkaline-earth atoms.
