# Optimal kernel estimates for a Schr\"odinger type operator

Research paper by **Anna Canale, Cristian Tacelli**

Indexed on: **13 Apr '16**Published on: **13 Apr '16**Published in: **Mathematics - Analysis of PDEs**

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#### Abstract

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