We consider the Schr\"odinger type operator ${ \mathcal A}=(1+|x|^{ \alpha})\Delta-|x|^{ \beta}$, for $\alpha\in [0,2]$ and $\beta\ge 0$. 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 $\beta\ge 2$, 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.
Categories: math.AP