Analytical solution of Mori's equation with hyperbolic secant memory

The equation of motion of the auto-correlation function has been solved analytically using a hyperbolic secant form of the memory function. The analytical result obtained for long-time expansion together with short-time expansion provides a good description over the whole time domain as judged by a comparison with the numerical solution of the Mori equation of motion. We also find that the time evolution of the auto-correlation function is determined by a single parameter tau which is related to frequency sum rules up to fourth order. The autocorrelation function has been found to show simple decaying or oscillatory behaviour depending on whether the parameter tau is greater than or less than some critical value. Similarities as well as differences in the time evolution of the auto-correlation have been discussed for exponential, hyperbolic secant and Gaussian approaches of the memory function