Abstract
We characterize functions \(V\le 0\) for which the heat kernel of the Schrödinger 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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Acknowledgements
We thank the referee for insightful comments and suggestions, which largely shaped the paper. In particular the paper merges the results of two preprints [3, 4], and the proof of Theorem 1.4 is much shorter than the elementary arguments given in [4].
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Krzysztof Bogdan was supported by the Polish National Science Center (Narodowe Centrum Nauki, NCN) Grant 2014/14/M/ST1/00600. Jacek Dziubański was supported by the NCN Grant DEC-2012/05/B/ST1/00672. Karol Szczypkowski was partially supported by IP2012 018472 and by the German Science Foundation (SFB 701).
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Bogdan, K., Dziubański, J. & Szczypkowski, K. Sharp Gaussian Estimates for Heat Kernels of Schrödinger Operators. Integr. Equ. Oper. Theory 91, 3 (2019). https://doi.org/10.1007/s00020-019-2501-y
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DOI: https://doi.org/10.1007/s00020-019-2501-y