  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.
