Dynamics of the independent-oscillator (IO) model in the Einstein and pseudo-Debye limits

The independent-oscillator model is thought to be a simple model for Brownian motion, explored extensively by Ford, Lewis and O'Connell. This model does not specify the distribution of the oscillators’ frequencies. By considering the distribution to be of two standard limits, otherwise preserving the Hermiticity of the model, we obtain the memory function exactly by the method of recurrence relations. In one limit the memory function is purely oscillatory while in the other limit it shows a decreasing amplitude with an oscillation. Relevance to transport theory is briefly pointed out.