  We prove that the heat kernel associated to the Schr\"odinger type operator
$A:=(1+|x|^\alpha)\Delta-|x|^\beta$ satisfies the estimate $$k(t,x,y)\leq
c_1e^{\lambda_0t}e^{c_2t^{-b}}\frac{(|x||y|)^{-\frac{N-1}{2}-\frac{\beta-\alpha}{4}}}{1+|y|^\alpha}
e^{-\frac{2}{\beta-\alpha+2}|x|^{\frac{\beta-\alpha+2}{2}}}
e^{-\frac{2}{\beta-\alpha+2}|y|^{\frac{\beta-\alpha+2}{2}}} $$ for
$t>0,|x|,|y|\ge 1$, where $c_1,c_2$ are positive constants and
$b=\frac{\beta-\alpha+2}{\beta+\alpha-2}$ provided that $N>2,\,\alpha\geq 2$
and $\beta>\alpha-2$. We also obtain an estimate of the eigenfunctions of $A$.
