On Schroedinger type operators with unbounded coefficients: Generation and heat kernel estimates

We consider the Schroedinger type operator ${mathcal A}=(1+|x|^{alpha})Delta-|x|^{eta}$, for $alpha in [0,2]$ and $eta ge 0$. We prove that, for any $p in (1,infty)$, the minimal realization of operator ${mathcal A}$ in $L^p(R^N)$ generates a strongly continuous analytic semigroup $(T_p(t))_{t ge 0}$. For $alpha in [0,2)$ and $eta ge 2$, we then prove some upper estimates for the heat kernel $k$ associated to the semigroup $(T_p(t))_{t ge 0}$. As a consequence we obtain an estimate for large $|x|$ of the eigenfunctions of ${mathcal A}$. Finally, we extend such estimates to a class of divergence type elliptic operators.