Elliptic operators with unbounded diffusion,drift and potential terms

We prove that the realization $A_p$ in $L^p({\mathbb{R}}^N),1<p<\infty$, of the elliptic operator $A=(1+|x{|}^{\alpha })\mathrm{\Delta }+b|x{|}^{\alpha ?1}\frac{x}{|x|}?\mathrm{?}?c|x{|}^{\beta }$ 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 $T(?)$ provided that $\alpha >2,\beta >\alpha ?2$ and any constants $b\in \mathbb{R}$ and $c>0$. This generalizes the recent results in 4] and in 16]. Moreover we show that $T(?)$ is consistent, immediately compact and ultracontractive.