Schrödinger type operators with unbounded diffusion and potential terms

Abstract

We prove that the realization $A_p$ in $L^p({\mathbb{R}}^N),1<p<\infty$, of the Schrödinger 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.