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We derive an analytical expression for the intermediate scattering functionof a particle on a flat surface obeying the Generalised Langevin Equation, withexponential memory friction. Numerical simulations based on an extended phasespace method confirm the analytical results. The simulated trajectories providequalitative insight into the effect that introducing a finite memory timescalehas on the analytical line shapes. The relative amplitude of the long-timeexponential tail of the line shape is suppressed, but its decay rate isunchanged, reflecting the fact that the cutoff frequency of the exponentialkernel affects short-time correlations but not the diffusion coefficient whichis defined in terms of a long-time limit. The exponential sensitivity of therelative amplitudes to the decay time of the chosen memory kernel is a verystrong indicator for the prospect of inferring a friction kernel and thephysical insights from experimentally measured intermediate scatteringfunctions.

The intermediate scattering function for quasi-elastic scattering in the presence of memory friction