Numerical Study of thetMatrix in the Kondo Problem

A numerical study of the Kondo problem is presented. The calculations are based on the Suhl-Abrikosov-Nagaoka integral equation for the scattering amplitude t(ω,T) of the s-d exchange Hamiltonian. Use is made of the exact analytic solution first given in detail by Zittartz and M\"uller-Hartmann. It is shown that, because of the resonance in Imt(ω,T) which occurs at the Fermi energy ω=0 at low temperatures, the tunneling density of states of a dilute paramagnetic alloy is very slightly reduced at zero bias voltage. There is, however, a possibility of detecting this change by studying the derivative of the conductance. The details of the Zittartz-M\"uller-Hartmann expression are found to be unimportant for the low-temperature behavior of the transport coefficients with the exception of the thermoelectric power. A reinvestigation of the thermoelectric power shows that some care is necessary in the evaluation of the integrals ${K}ₙ={∫}{{-}{∞}}^{{∞}}d{ω}{{ω}}ⁿ{n}₀({ω}){τ}({ω},T){{∂}f}{{∂}{ω}}$, because of the strong ${ω}$ dependence of the electronic lifetime ${τ}({ω},T){∝}{[Imt({ω},T)]}^{{-}1}$ at low temperatures. While the transport coefficients reflect the behavior of the scattering amplitude in a small energy interval about the Fermi energy, the specific-heat anomaly is found to be related to the temperature derivative of t(ω,T) at large values of the energy variable ω comparable to the bandwidth. We also point out that the quasiparticle approximation is not valid for electrons interacting with impurity spins due to the rapid variation of t(ω,T) near the Fermi energy.