The authors study the spectral properties of a localized particle (a deep core level, or a heavy particle hopping in a solid), coupled to conduction electrons that are described by a marginal Fermi liquid hypothesis. The main result is that, in this model, the core level line shape in an X-ray photoemission experiment shifts with temperature. The spectrum is consistent with the decrease of the hopping rate with decreasing temperature obtained by Zhang et al. (1992). In the presence of a distribution of energy levels for the hopping particle, the temperature dependence of the hopping rate is found to be less pronounced. The observability of both the shift in the X-ray photoemission spectrum and the decrease of the hopping rate depends on the strength of the interaction. A rough estimate of the value of the interaction parameter indicates that the effect might be observable in X-ray photoemission experiments.