  We prove that the realization $A_p$ in $L^p(\mathbb{R}^N),\,1<p<\infty$, of
the elliptic operator
$A=(1+|x|^{\alpha})\Delta+b|x|^{\alpha-1}\frac{x}{|x|}\cdot
\nabla-c|x|^{\beta}$ with domain $D(A_p) =\{ u \in W^{2,p}(\mathbb{R}^N)\, |\,
Au \in L^p(\mathbb{R}^N)\}$ generates a strongly continuous analytic semigroup
$T(\cdot)$ provided that $\alpha >2,\,\beta >\alpha -2$ and any constants $b\in
\mathbb{R}$ and $c>0$. This generalizes the recent results in [A.Canale, A.
Rhandi, C. Tacelli, Ann. Sc. Norm. Super. Pisa CI. Sci. (5), 2016] and in
[G.Metafune, C.Spina, C.Tacelli, Adv. Diff. Equat., 2014]. Moreover we show
that $T(\cdot)$ is consistent, immediately compact and ultracontractive.
