  In this paper, we derive a series of exact analytical closed expressions to
calculate the complex eigenfrequencies and the displacement for the
corresponding eigenmodes of a viscoelastic (nano-)sphere in the presence of
linear damping. Where possible, we provide closed expressions for damping
rates, including the contributions from viscosity, as well as thermal
conductivity and solutions of the heat equation. We assume an isolated system,
such that no energy/heat transfer to the environment is allowed. We find
monotonic behavior of the damping as a function of frequency for breathing and
torsional modes, however, for spheroidal modes we find non-monotonicity.
Furthermore, we analytically analyze the thermodynamic limit for all mode
types. We also investigate the frequency shift and find expected behavior, i.e.
a reduced eigenfrequency with damping than without damping for breathing and
torsional modes. For spheroidal modes, however, we find non-monotonic shifts,
corresponding to the damping. For some eigenfrequencies we find anomalous
frequency shifts.
