Fermi-edge singularity in the vicinity of the resonant scattering condition

Fermi-edge absorption theory predicting the spectrum, A(\omega)\propto \omega^{-2\delta_0/\pi+\delta^2_0/\pi^2}, relies on the assumption that scattering phase, \delta_0, is frequency-independent. Dependence of \delta_0 on \omega becomes crucial near the resonant condition, where the phase changes abruptly by \pi. In this limit, due to finite time spent by electron on a resonant level, the scattering is dynamic. We incorporate this time delay into the theory, solve the Dyson equation with a modified kernel and find that, near the resonance, A(\omega) behaves as \omega^{-3/4} |\ln \omega|. Resonant scattering off the core hole takes place in 1D and 2D in the presence of an empty subband above the Fermi level; then attraction to hole splits off a resonant level from the bottom of the empty subband. Fermi-edge absorption in the regime when resonant level transforms into a Kondo peak is discussed.