Optimal kernel estimates for a Schrödinger type operator

In the paper the principal result obtained is the estimate for the heat kernel associated to the Schrödinger type operator $(1+|x|^α)Δ-|x|^β$ $k(t,x,y)\leq Ct^{-\fracθ{2}}\frac {φ(x)φ(y)}{1+|x|^α},$ where $φ=(1+|x|^α)^{\frac{2-θ}{4}+\frac{1}α\frac{θ-N}{2}}$, $θ\geq N$ and $0<t<1$, provided that $N>2$, $α> 2$ and $β>α-2$. This estimate improves a similar estimate in \cite {can-rhan-tac2} with respect to the dependence on spatial component.

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