Optimal kernel estimates for a Schr\"odinger type operator

TL;DR

This paper derives improved heat kernel estimates for a Schrödinger-type operator with specific potential growth conditions, enhancing previous bounds by refining spatial dependence for certain parameter ranges.

Contribution

It provides new optimal kernel estimates for a Schrödinger operator with polynomial potentials, improving upon prior bounds by refining spatial dependence.

Findings

Derived explicit heat kernel bounds for the operator.

Extended the parameter range where estimates are valid.

Improved spatial dependence in kernel estimates.

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.