Summary: We prove that the heat kernel associated to the Schrödinger type operator \(A:=(1+|x|^\alpha)\Delta-|x|^\beta\) satisfies the estimate \[ k(t,x,y)\leq c_1e^{\lambda_0t}e^{c_2t^{-b}}\frac{(|x||y|)^{-\frac{N-1}{2}-\frac{\beta-\alpha}{4}}}{1+|y|^\alpha} e^{-\frac{\sqrt{2}}{\beta-\alpha+2}|x|^{\frac{\beta-\alpha+2}{2}}} e^{-\frac{\sqrt{2}}{\beta-\alpha+2}|y|^{\frac{\beta-\alpha+2}{2}}} \] for \(t>0,|x|,|y|\geq 1\), 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 \(A\).