  Simulations and theory are presented for the power spectral density functions
(PSDs) of particles in time dependent and anharmonic potentials including the
effects of a thermal environment leading to damping and fluctuating forces. We
investigate three one dimensional perturbations to the harmonic oscillator of
which two are time dependent changes in the natural frequency of the
oscillator, while the other is a time independent extension of the quadratic
potential to include a quartic term. We investigate the effect of these
perturbations on two PSDs of the motion that are used in experiments on trapped
nano-oscillators. We also derive and numerically test the PSDs for the motion
of a spherical nanoparticle in a Paul trap. We found that the simple harmonic
Langevin oscillator's PSDs are good approximations for the $x$-and
$y$-coordinates' PSDs for small values of the parameter $q$ of the Mathieu
equation, but the difference can be more than a factor of two as '$q$'
increases. We also numerically showed that the presence of a permanent electric
dipole on the nanosphere can significantly affect the PSDs in the $x$-and
$y$-coordinates.
