Heat kernel for fractional Schr\"{o}dinger operators with unbounded potentials

：王健 （福建师范大学）
：2023-02-16 14:00
：厦大海韵园实验楼105报告厅

报告人：王健 （福建师范大学）

时  间：2023年2月16日下午14:00-14:40

地  点：厦大海韵园实验楼105报告厅

内容摘要:

We establish global in time and qualitatively sharp bounds for the heat kernel of  the fractional Schr\"odinger operator $\mathcal L^V=-(-\Delta)^{\alpha/2}+V$ on $\R^d$, where $\alpha\in (0,2)$ and $V:\R^d\to [0,\infty)$ is a nonnegative and locally bounded potential on $\R^d$ so that for all $x\in \R^d$ with $|x|\ge 1$, $c_1g(|x|)\le V(x)\le c_2g(|x|)$ with some constants $c_1,c_2>0$ and a nondecreasing and strictly positive function $g:[0,\infty)\to [1,+\infty)$ that satisfies $g(2r)\le c_0 g(r)$ for all $r>0$ and $\lim_{r\to \infty} g(r)=\infty.$ The approach is mainly based on probabilistic tools including the L\'evy system of jump processes, with aid of novel iteration arguments due to the heavy-tailed property of symmetric $\alpha$-stable processes. In particular, we can also present global in space and time two-sided bounds of heat kernel even when the associated Schr\"{o}dinger semigroup is not intrinsically ultracontractive.  

个人简介：

王健，福建师范大学数学与统计学院教授、博士生导师、国家杰出青年基金获得者。 主要从事随机过程与随机分析方向的研究，特别是Lévy型过程的随机分析。