Golden-rule approach to the soft-x-ray-absorption problem. III. The temperature dependence

The exact formulation of the soft-x-ray-absorption problem of previous work [Phys. Rev. B 28, 6833 (1983)] is generalized to the case of finite temperatures. By giving a closed-form expression of the response function in terms of the overlap integrals between the initial and final electron states, we demonstrate that the exactness can be pursued quite generally. The general formula, when applied to the case of a contact core-hole potential, leads to expressions of the dispersion integral and the frequency shift, both modified from their $T=0$ forms by the temperature effect. In the nearedge region, an analytical form of the response function can be derived, which reproduces precisely the time and temperature dependences of that of Yuval and Anderson [Phys. Rev. B 1, 1522 (1970)] and, in addition, provides an exact expression for the prefactor of the open-line contribution of the spectrum.