Green's function of the core electrons in a lattice model for x-ray spectra: Breakdown of perturbation theory in the long-time limit.

Abstract

A model for the description of the x-ray spectra of an infinitely heavy particle embedded in a sea of conduction electrons is extended to the case of many, randomly distributed local core electrons. The mean-field thermodynamics, i.e., the exact solution in d=\ensuremath{\infty}, is presented and the Green's function of the heavy electrons is thoroughly studied. The exact asymptotic limits of this function for short and long, imaginary and real times are obtained for all interaction strengths. Two important observations are made. First, the zero-temperature function exhibits for long imaginary and real times the edge singularity independently of the ground state of the conduction electrons. Second, the finite-time approximate schemes fail to reproduce the exact long-time asymptotics at any temperature and the Green's function of the local electrons undergoes a nonanalytic crossover from short to long times. The exact long-time asymptotics are available only via the Wiener-Hopf or related nonperturbative techniques.