  We consider the rates of relaxation of a particle in a harmonic well, subject
to L\'evy noise characterized by its L\'evy index $\mu$. Using the propagator
for this L\'evy Ornstein-Uhlenbeck process (LOUP), we show that the eigenvalue
spectrum of the associated Fokker-Planck operator has the form $(n+m\mu)\nu$
where $\nu$ is the force constant characterizing the well, and
$n,m\in\mathbb{N}$. If $\mu$ is irrational, the eigenvalues are all
non-degenerate, but rational $\mu$ can lead to degeneracy. The maximum
degeneracy is shown to be two. The left eigenfunctions of the fractional
Fokker-Planck operator are very simple while the right eigenfunctions may be
obtained from the lowest eigenfunction by a combination of two different
step-up operators. Further, we find that the acceptable eigenfunctions should
have the asymptotic behavior $|x|^{-n_1+n_2\;\mu}$ as $|x| \rightarrow \infty$,
with $n_1$ and $n_2$ being positive integers, though this condition alone is
not enough to identify them uniquely. We also assert that the rates of
relaxation of LOUP are determined by the eigenvalues of the associated
fractional Fokker-Planck operator and do not depend on the initial state if the
moments of the initial distribution are all finite. If the initial distribution
has fat tails, for which the higher moments diverge, one would have
non-spectral relaxation, as pointed out by Toenjes et. al (Physical Review
Letters, 110, 150602 (2013)).

Statistical Mechanics