Schr\"odinger type operators with unbounded diffusion and potential terms

We prove that the realization $A_p$ in $L^p(\mathbb{R}^N),\,1<p<\infty$, of the Schr\"odinger type operator $A=(1+|x|^{\alpha})\Delta-|x|^{\beta}$ 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 provided that $N>2,\,\alpha >2$ and $\beta >\alpha -2$. Moreover this semigroup is consistent, irreducible, immediately compact and ultracontractive.