A generalized Anderson model for a magnetic impurity in an interacting one-dimensional electron gas is studied via a mapping onto a classical Coulomb gas. For weak potential scattering, the local-moment parameter regime expands as repulsive bulk interactions become stronger, but the Kondo scale for the quenching of the impurity moment varies non-monotonically. There also exist two regimes dominated by backward potential scattering: one in which the impurity is non-magnetic, and another in which an unquenched local moment survives down to very low temperatures.
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(\ensuremath{\epsilon})\ensuremath{\propto}${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$, with \ensuremath{\alpha}=3/8+${\mathrm{\ensuremath{\delta}}}_{+}$/\ensuremath{\pi}-${\mathrm{\ensuremath{\delta}}}_{+}^{2}$/2${\mathrm{\ensuremath{\pi}}}^{2}$, where ${\mathrm{\ensuremath{\delta}}}_{+}$ is the forward-scattering phase shift of the core hole. In contrast to previous theories, \ensuremath{\alpha} is finite (and universal) in the limit of weak core-hole potential. In the case of weak backscattering U(2
kF), the exponent in the power-law dependence of absorption on energy crosses over to a value \ensuremath{\alpha}=${\mathrm{\ensuremath{\delta}}}_{+}$/\ensuremath{\pi}-${\mathrm{\ensuremath{\delta}}}_{+}^{2}$/2${\mathrm{\ensuremath{\pi}}}^{2}$ above an energy scale ${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{*}}$\ensuremath{\sim}[U(2
kF)${]}^{1/\ensuremath{\gamma}}$, where \ensuremath{\gamma} is a dimensionless measure of the electron-electron interactions.
Scattering from a localized potential of arbitrary strength is found for weakly interacting electrons. Poor man's renormalization group procedure allows us to calculate the renormalized transmission amplitudes at any energies. Simple formulas that describe the conductance at any temperature are derived. In the presence of the electron-electron backscattering, the low-temperature conductance deviates from the results of Luttinger liquid theory. In particular, the temperature dependence of the conductance may become nonmonotonic. In the presence of a magnetic field, backscattering gives rise to a peak in the differential conductance at bias equal to the Zeeman splitting.