Optimal kernel estimates for a Schrödinger type operator

Abstract

In the paper the principal result obtained is the estimate for the heat kernel associated to the Schrödinger type operator $(1+|x{|}^{\alpha })\Delta -|x{|}^{\beta }$ \[ k(t,x,y)\leq Ct^{-\fracθ{2}}\frac {φ(x)φ(y)}{1+|x|^α}, \] where $\phi =(1+|x{|}^{\alpha }{)}^{\frac{2-\theta }{4}+\frac{1}{\alpha }\frac{\theta -N}{2}}$, $\theta \ge N$ 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.