Optimal kernel estimates for a Schr\"odinger type operator

In the paper the principal result obtained is the estimate for the heat kernel associated to the Schr"odinger type operator $(1+\mathrm{\mid }x{\mathrm{\mid }}^{\alpha })\mathrm{\Delta }-\mathrm{\mid }x{\mathrm{\mid }}^{\beta }$ $$k(t,x,y)\le C{t}^{-\frac{\theta }{2}}\frac{\phi (x)\phi (y)}{1+\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }},$$ where $\phi =(1+\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }{)}^{\frac{2-\theta }{4}+\frac{1}{\alpha }\frac{\theta -N}{2}}$, $\theta \ge N$ and $0<t<1$, provided that $N>2$, $\alpha >2$ and $\beta >\alpha -2$. This estimate improves a similar estimate in \cite {can-rhan-tac2} with respect to the dependence on spatial component.