Fermi Edge Singularities and Backscattering in a Weakly Interacting 1D Electron Gas

The photon-absorption edge in a weakly interacting one-dimensional electron gas is studied, treating backscattering of conduction electrons from the core hole exactly. Close to threshold, there is a power-law singularity in the absorption, I(ǫ) ∝ ǫ−α, with α = 3/8 + δ+/π − δ2 +/2π where δ+ is the forward scattering phase shift of the core hole. In contrast to previous theories, α is finite (and universal) in the limit of weak core hole potential. In the case of weak backscattering U(2kF ), the exponent in the power-law dependence of absorption on energy crosses over to a value α = δ+/π − δ2 +/2π above an energy scale ǫ ∼ [U(2kF )]1/γ , where γ is a dimensionless measure of the electron-electron interactions. PACS numbers: 78.70.Dm, 73.20.Dx Typeset using REVTEX