  Exact expressions are derived for the intermediate scattering function (ISF)
of a quantum particle diffusing in a harmonic potential and linearly coupled to
a harmonic bath. The results are valid for arbitrary strength and spectral
density of the coupling. The general, exact non-Markovian result is expressed
in terms of the classical velocity autocorrelation function, which represents
an accumulated phase during a scattering event. The imaginary part of the
exponent of the ISF is proportional to the accumulated phase, which is an
antisymmetric function of the correlation time $t$. The expressions extend
previous results given in the quantum Langevin framework where the classical
response of the bath was taken as Markovian. For a special case of
non-Markovian friction, where the friction kernel decays exponentially in time
rather than instantaneously, we provide exact results relating to unconfined
quantum diffusion, and identify general features that allow insight to be
exported to more complex examples. The accumulated phase as a function of the t
has a universal gradient at the origin, depending only on the mass of the
diffusing system particle. At large t the accumulated phase reaches a constant
limit that depends only on the classical diffusion coefficient and is therefore
independent of the detailed memory properties of the friction kernel.
Non-Markovian properties of the friction kernel are encoded in the details of
how the accumulated phase switches from its $t\rightarrow -\infty$ to its
$t\rightarrow -\infty$ limit, subject to the constraint of the universal
gradient. When memory effects are significant, the transition from one limit to
the other becomes non-monotonic, owing to oscillations in the classical
velocity autocorrelation. The result is interpreted in terms of a solvent
caging effect, in which slowly fluctuating bath modes create transient wells
for the system particle.
