Complete Wiener-Hopf Solution of the X-Ray Edge Problem

We present a complete solution of the soft x-ray edge problem within a field-theoretic approach based on the Wiener-Hopf infinite-time technique. We derive for the first time within this approach critical asymptotics of all the relevant quantities for the x-ray problem as well as their nonuniversal prefactors. Thereby we obtain the most complete field-theoretic solution of the problem with a number of new experimentally relevant results. We make thorough comparison of the proposed Wiener-Hopf technique with other approaches based on finite-time methods. It is proven that the Fredholm, finite-time solution converges smoothly to the Wiener-Hopf one and that the latter is stable with respect to perturbations in the long-time limit. Further on we disclose a wide interval of intermediate times showing quasicritical behavior deviating from the Wiener-Hopf one. The quasicritical behavior of the core-hole Green function is derived exactly from the Wiener-Hopf solution and the quasicritical exponent is shown to match the result of Nozi\`eres and De Dominicis. The reasons for the quasicritical behavior and the way of a crossover to the infinite-time solution are expounded and the physical relevance of the Nozi\`eres and De Dominicis as well as of the Winer-Hopf results are discussed.