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

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

Indexed on: **05 Jan '15**Published on: **05 Jan '15**Published in: **Mathematics - Analysis of PDEs**

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#### Abstract

We prove that the heat kernel associated to the Schr\"odinger type operator
$(1+|x|^\alpha)\Delta-|x|^\beta$ satisfies the estimate $$k(t,x,y)\leq
c_1e^{\lambda_0t}e^{c_2t^{-b}}\frac{(|x||y|)^{-\frac{N-1}{2}-\frac{\beta-\alpha}{4}}}{1+|y|^\alpha}
e^{-\int_1^{|x|}\frac{s^{\beta/2}}{\sqrt{1+s^\alpha}}\,ds}
e^{-\int_1^{|y|}\frac{s^{\beta/2}}{\sqrt{1+s^\alpha}}\,ds} $$ for
$t>0,|x|,|y|\ge 1$, where $c_1,c_2$ are positive constants and
$b=\frac{\beta-\alpha+2}{\beta+\alpha-2}$ provided that for $N>2,\,\alpha\geq
2$ and $\beta>\alpha-2$. We also obtain an estimate of the eigenfunctions of
$A$.