The free electron Kondo problem can be described by a one-dimensional (1D)
model because only the s-wave part of the electronic wave-function is affected
by the Kondo coupling. Since only the spin degrees of freedom are involved in
the Kondo interaction, and due to spin-charge separation in 1D, the universal
low energy long distance physics of the Kondo model also arises when a magnetic
impurity is coupled to the end of a gap-less antiferromagnetic J_1-J_2 spin-1/2
chain, where J_1(J_2) is the (next-)nearest neighbor coupling. Experimental
realizations of such spin chain models are possible and using various
analytical and numerical techniques, we present a detailed and quantitative
comparison between the usual free electron Kondo model and such spin chain
versions of the Kondo problem. For the gap-less J_1-J_2 spin chain two cases
are studied, with zero next nearest neighbor coupling, J_2=0, and with a
critical second neighbor coupling, J_2=J_2c. We first focus on the spin chain
impurity model at J_2c~0.2412 where a bulk marginal coupling present in the
spin chain model for J_2