  The properties of a dissipative system depend on the spectral density of the
coupling to the environment. Mostly, the dependence on the low-frequency
behavior is in the focus of interest. However, in order to avoid divergencies,
it is also necessary to suppress the spectral density of the coupling at high
frequencies. Interestingly, the very existence of this cutoff may lead to a
mass renormalization which can have drastic consequences for the thermodynamic
properties of the dissipative system. Here, we explore the role which the
cutoff in the spectral density of the coupling plays for a free damped particle
and we compare the effect of an algebraic cutoff with that of a sharp cutoff.
