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arXiv · 1706.06172

Sharp Gaussian estimates for heat kernels of Schr\"odinger operators

Krzysztof Bogdan·Jacek Dziubański·Karol Szczypkowski
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Abstract

We characterize functions $V\le 0$ for which the heat kernel of the Schr\"o\-dinger operator $\Delta+V$ 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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Krzysztof Bogdan, Jacek Dziubański, Karol Szczypkowski. 2017-06-19. Sharp Gaussian estimates for heat kernels of Schr\"odinger operators. https://arxiv.org/abs/1706.06172

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