  We consider the generalized Langevin equation (GLE) in a harmonic potential
with power law decay memory. We study the anomalous diffusion of the particle's
displacement and velocity. By comparison with the free particle situation in
which the velocity was previously shown to be either diffusive or subdiffusive,
we find that, when trapped in a harmonic potential, the particle's displacement
may either be diffusive or superdiffusive. Under slightly stronger assumptions
on the memory kernel, namely, for kernels related to the broad class of
completely monotonic functions, we show that both the free particle and the
harmonically bounded GLE satisfy the equipartition of energy condition. This
generalizes previously known results for the GLE under particular kernel
instances such as the generalized Rouse kernel or (exactly) a power law
function.
