  Let $L$ be a linear symmetric differential operators on $L^{2}\left(
\mathbb{R}\right) $ whose domain is the Schwartz test function space,
$\mathcal{S}.$ For the majority of this paper, it is assumed that the
coefficient of $L$ are polynomial functions on $\mathbb{R}.$ We will give
criteria on the polynomial coefficients of $L$ which guarantees that $L$ is
essentially self-adjoint, $\bar{L}\geq-CI$ for some $C<\infty,$ and that
$\mathcal{S}$ is a core for $\left( \bar{L}+C\right) ^{r}$ for all $r\geq0.$
Given another polynomial coefficient differential operator, $\tilde{L},$ we
will further give criteria on the coefficients $L$ and $\tilde{L}$ which
implies operator comparison inequalities of the form $\left( \overline
{\tilde{L}}+\tilde{C}\right) ^{r}\leq C_{r}\left( \bar{L}+C\right) ^{r}$ for
all $0\leq r<\infty.$ The last inequality generalized to allow for an added
parameter, $\hbar>0,$ in the coefficients is used to provide a large class of
operators satisfying the hypotheses in our another paper "On the classical
limit of quantum mechanics" (will be submitted very soon) where a strong form
of the classical limit of quantum mechanics is shown to hold.
