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We investigate the momentum distribution function of a single distinguishableimpurity particle which formed a polaron state in a gas of either free fermionsor Tonks-Girardeau bosons in one spatial dimension. We obtain a Fredholmdeterminant representation of the distribution function for the Bethe ansatzsolvable model of an impurity-gas $\delta$-function interaction potential atzero temperature, in both repulsive and attractive regimes. We deduce from thisrepresentation the fourth power decay at a large momentum, and a weaklydivergent (quasi-condensate) peak at a finite momentum. We also demonstratethat the momentum distribution function in the limiting case of infinitelystrong interaction can be expressed through a correlation function of theone-dimensional impenetrable anyons.

Zero temperature momentum distribution of an impurity in a polaron state of one-dimensional Fermi and Tonks-Girardeau gases