Resonance in One--Dimensional Fermi--Edge Singularity

The problem of the Fermi--edge singularity in a one--dimensional Tomonaga--Luttinger liquid is reconsidered. The backward scattering of the conduction band electrons on the impurity--like hole in the valence band is analyzed by mapping the problem onto a Coulomb gas theory. For the case when the electron--electron interaction is repulsive the obtained exponent of the one--dimensional Fermi--edge singularity appears to be different from the exponent found in the previous studies. It is shown that the infrared physics of the Fermi--edge singularity in the presence of backward scattering and electron--electron repulsion resembles the physics of the Kondo problem.