Schrödinger 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 Schrödinger type operator $A=(1+|x|^α)Δ-|x|^β$ 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 $N>2,\,α>2$ and $β>α-2$. Moreover this semigroup is consistent, irreducible, immediately compact and ultracontractive.

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