Dispersion relation and spectral function of an impurity in a one-dimensional quantum liquid

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We consider the motion of an impurity particle in a general one-dimensional quantum fluid at zero temperature. The dispersion relation Ω(P) of the impurity is strongly affected by interactions with the fluid as the momentum approaches ±πℏ n, ± 3πℏ n,..., where n is the density. This behavior is caused by singular ± 2πℏ n scattering processes and can be understood by analogy to the Kondo effect, both at strong and weak coupling, with the possibility of a quantum phase transition where Ω'(± πn) jumps to zero with increasing coupling. The low energy singularities in the impurity spectral function can be understood on the same footing.