Fermi-edge singularities and backscattering in a weakly interacting one-dimensional 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(ε)∝${\mathrm{\epsilon }}^{\mathrm{-}\mathrm{\alpha }}$, with α=3/8+${\mathrm{\delta }}_{+}$/π-${\mathrm{\delta }}_{+}^2$/2${\mathrm{\pi }}^2$, where ${\mathrm{\delta }}_{+}$ 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(2${\mathit{k}}_{\mathit{F}}$), the exponent in the power-law dependence of absorption on energy crosses over to a value α=${\mathrm{\delta }}_{+}$/π-${\mathrm{\delta }}_{+}^2$/2${\mathrm{\pi }}^2$ above an energy scale ${\mathrm{\epsilon }}^{\mathrm{*}}$∼[U(2${\mathit{k}}_{\mathit{F}}$)${]}^{1/\gamma }$, where γ is a dimensionless measure of the electron-electron interactions.