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

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 $\pmπ\hbar n, \pm 3π\hbar n, ...$, where $n$ is the density. This behavior is caused by singular $\pm 2π\hbar n$ 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 $Ω'(\pm πn)$ jumps to zero with increasing coupling. The low energy singularities in the impurity spectral function can be understood on the same footing.