Threshold bound states and zero modes of fermions in a selfdual Chern-Simons vortex background

The existence of bound states at the mass threshold and zero modes for fermions interacting with a self-dual Chern-Simons vortex (i.e., a classical solution of the Chern-Simons Abelian Higgs model without the Maxwell term) is investigated. We develop a procedure that allows one to find all the regular solutions of the interacting Dirac equation at the mass threshold. We show, on one hand, that in the minimal coupling zero modes and bound states at threshold do not exist and, on the other hand, that, by adding to the fermion Lagrangian two different types of Higgs-fermion interactions, such modes can be generated for specific values of the coupling constants.