Sharp Gaussian estimates for heat kernels of Schrödinger operators

2017/06/19 by Bogdan, Krzysztof, Dziubański, Jacek, Szczypkowski, Karol
#35A08 #35B25 (Secondary) #47D06 #47D08 (Primary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1706.06172

Abstract

We characterize functions V≤ 0 for which the heat kernel of the Schrö\-dinger operator Δ+V is comparable with the Gauss-Weierstrass kernel uniformly in space and time. In dimension 4 and higher the condition turns out to be more restrictive than the condition of the boundedness of the Newtonian potential of V. This resolves the question of V.~Liskevich and Y.~Semenov posed in 1998. We also give specialized sufficient conditions for the comparability, showing that local L^p integrability of V for p>1 is not necessary for the comparability.

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