RENORMALIZATION OF POTENTIAL SCATTERING IN A ONE-DIMENSIONAL NON-LUTTINGER LIQUID - CONDUCTION AND X-RAY FERMI-EDGE SINGULARITY

We review some of our recent results on the potential scattering in a weakly interacting one-dimensional(1D) electron gas. The technique we developed is a poor man's renormalization group procedure in the scatt ered wave basis. This technique can treat the renormalizations of the scattering on the barrier and the scattering between the electrons in a coherent way, and it allows us to find the scattering amplitudes on a localized potential of arbitrary strength for electrons at any energ y. The obtained phase shifts are used to study the Fermi-edge singular ity in an interacting 1D electron system, where anomalous exponent of the power-law singularity in the vicinity of the edge is found. The tr ansmission coefficient is directly related to the conductance of a 1D channel by the Landauer formula. Simple formulas that describe the con ductance at any temperature are derived. In spin-1/2 systems, the elec tron-electron backscattering induces renormalizations of the interacti on constants, which causes the low-temperature conductance to deviate from the results of the Luttinger liquid theory. In particular, the te mperature dependence of the conductance may become nonmonotonic. In th e presence of a magnetic field, backscattering gives rise to a peak in the differential conductance at bias equal to the Zeeman splitting.