Kernel estimates for Schr\"odinger type operators with unbounded diffusion and potential terms

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