Temperature Dependence of the Infrared Divergence in the Degenerate Fermi Gas

A sudden change in a short-range single-particle potential excites an infinite number of low-energy particlehole pairs in a degenerate Fermion gas. Although each order of perturbation theory diverges, the perturbation series can be summed formally. The sum agrees in the limit of zero temperature with a result due to Nozi\`eres and De Dominicis. At finite temperature, it confirms a conjecture of Anderson's. The excitation spectrum at energies that are small compared to the temperature can be interpreted in terms of a relaxation frequency, which is proportional to the temperature times the square of the phase shift.